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In [6]:
fi = interpolate.interp1d(x,func(x),kind='linear')
plot1(fi,label='Linear')
In [7]:
for knd, clr in ('previous','m'),('next','b'),('nearest','g'):
fi = interpolate.interp1d(x,func(x),kind=knd)
plot1(fi,label=knd,color=clr)
plt.show()
In [8]:
for knd, clr in ('slinear','m'),('quadratic','b'),('cubic','g'):
fi = interpolate.interp1d(x,func(x),kind=knd)
plot1(fi,color=clr,label=knd)
In [9]:
# Approximation errors
# x = np.sort(np.random.uniform(-5,10,11)) # generate new data
fi = interpolate.interp1d(x,func(x),kind=knd,bounds_error=False)
xd = np.linspace(-5,10,1000)
erd=np.abs(func(xd)-fi(xd))
plt.plot(xd,erd,color=clr)
print('Max error with %s splines is %1.5e'%(knd,np.nanmax(erd)))
Max error with slinear splines is 1.05142e+00
Max error with quadratic splines is 3.89974e-01
Max error with cubic splines is 4.35822e-01
In [10]:
# Approximation errors for regular grid
fi = interpolate.interp1d(xr,func(xr),kind=knd,bounds_error=False)
xd = np.linspace(-5,10,1000)
erd=np.abs(func(xd)-fi(xd))
plt.plot(xd,erd,color=clr)
print('Max error with %s splines is %1.5e'%(knd,np.nanmax(erd)))
Max error with slinear splines is 4.63043e-01
Max error with quadratic splines is 3.48546e-01
Max error with cubic splines is 1.89578e-01
#### Accuracy of the interpolation¶
How to reduce approximation errors?
• Number of nodes (more is better)
• Location of nodes (regular is better)
• Interpolation type (match function of interest)
In economic models we usually can control all of these
### Polynomial approximation/interpolation¶
Back to the beginning to explore the idea of replacing original $f(x)$ with simpler $g(x)$
• Data set $\{(x_i,f(x_i)\}, i=0,\dots,n$
• Functional form is polynomial of degree $n$ such that $g(x_i)=f(x_i)$
• If $x_i$ are distinct, coefficients of the polynomial are uniquely identified
Does polynomial $g(x)$ converge to $f(x)$ when there are more points? | {
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Does polynomial $g(x)$ converge to $f(x)$ when there are more points?
In [11]:
from numpy.polynomial import polynomial
degree = len(x)-1 # passing through all dots
p = polynomial.polyfit(x,func(x),degree)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,label='Polynomial of degree %d'%degree,extrapolation=True)
In [12]:
# now with regular grid
degree = len(x)-1 # passing through all dots
p = polynomial.polyfit(xr,func(xr),degree)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(xr,func(xr)),label='Polynomial of degree %d'%degree,extrapolation=True)
In [13]:
# how number of points affect the approximation (with degree=n-1)
for n, clr in (5,'m'),(10,'b'),(15,'g'),(25,'r'):
x2 = np.linspace(-5,10,n)
p = polynomial.polyfit(x2,func(x2),n-1)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(x2,func(x2)),label='%d points'%n,color=clr,extrapolation=True)
plt.show()
In [14]:
# how locations of points affect the approximation (with degree=n-1)
np.random.seed(2025)
n=8
for clr in 'b','g','c':
x2 = np.linspace(-4,9,n) + np.random.uniform(-1,1,n) # perturb points a little
p = polynomial.polyfit(x2,func(x2),n-1)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(x2,func(x2)),label='%d points'%n,color=clr,extrapolation=True)
plt.show()
In [15]:
# how degree of the polynomial affects the approximation
for degree, clr in (7,'b'),(9,'g'),(11,'m'):
p=polynomial.polyfit(xr,func(xr),degree)
fi=lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(xr,func(xr)),label='Polynomial of degree %d'%degree,color=clr,extrapolation=True)
#### Least squares approximation¶
We could also go back to function approximation and fit polynomials of lower degree
• Data set $\{(x_i,f(x_i)\}, i=0,\dots,n$
• Any functional form $g(x)$ from class $G$ that best approximates $f(x)$
$$g = \arg\min_{g \in G} \lVert f-g \rVert ^2$$
### Orthogonal polynomial approximation/interpolation¶
• Polynomials over domain $D$
• Weighting function $w(x)>0$
Inner product | {
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• Polynomials over domain $D$
• Weighting function $w(x)>0$
Inner product
$$\langle f,g \rangle = \int_D f(x)g(x)w(x)dx$$
$\{\phi_i\}$ is a family of orthogonal polynomials w.r.t. $w(x)$ iff
$$\langle \phi_i,\phi_j \rangle = 0, i\ne j$$
#### Best polynomial approximation in L2-norm¶
Let $\mathcal{P}_n$ denote the space of all polynomials of degree $n$ over $D$
$$\lVert f - p \rVert_2 = \inf_{q \in \mathcal{P}_n} \lVert f - q \rVert_2 = \inf_{q \in \mathcal{P}_n} \left[ \int_D ( f(x)-g(x) )^2 dx \right]^{\tfrac{1}{2}}$$
if and only if
$$\langle f-p,q \rangle = 0, \text{ for all } q \in \mathcal{P}_n$$
Orthogonal projection is the best approximating polynomial in L2-norm
#### Uniform (infinity, sup-) norm¶
$$\lVert f(x) - g(x) \rVert_{\infty} = \sup_{x \in D} | f(x) - g(x) | = \lim_{n \rightarrow \infty} \left[ \int_D ( f(x)-g(x) )^n dx \right]^{\tfrac{1}{n}}$$
Measures the absolute difference over the whole domain $D$
#### Chebyshev (minmax) approximation¶
What is the best polynomial approximation in the uniform (infinity, sup) norm?
$$\lVert f - p \rVert_{\infty} = \inf_{q \in \mathcal{P}_n} \lVert f - q \rVert_{\infty} = \inf_{q \in \mathcal{P}_n} \sup_{x \in D} | f(x) - g(x) |$$
Chebyshev proved existence and uniqueness of the best approximating polynomial in uniform norm.
#### Chebyshev polynomials¶
• $[a,b] = [-1,1]$ and $w(x)=(1-x^2)^{(-1/2)}$
• $T_n(x)=\cos\big(n\cos^{-1}(x)\big)$
• Recursive formulas:
$$\begin{eqnarray} T_0(x)=1,\\ T_1(x)=x,\\ T_{n+1}(x)=2x T_n(x) - T_{n-1}(x) \end{eqnarray}$$
#### Accuracy of Chebyshev approximation¶
Suppose $f: [-1,1]\rightarrow R$ is $C^k$ function for some $k\ge 1$, and let $I_n$ be the degree $n$ polynomial interpolation of $f$ with nodes at zeros of $T_{n+1}(x)$. Then
$$\lVert f - I_n \rVert_{\infty} \le \left( \frac{2}{\pi} \log(n+1) +1 \right) \frac{(n-k)!}{n!}\left(\frac{\pi}{2}\right)^k \lVert f^{(k)}\rVert_{\infty}$$
📖 Judd (1988) Numerical Methods in Economics | {
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📖 Judd (1988) Numerical Methods in Economics
• achieves best polynomial approximation in uniform norm
• works for smooth functions
• easy to compute
• but does not approximate $f'(x)$ well
#### General interval¶
• Not hard to adapt the polynomials for the general interval $[a,b]$ through linear change of variable
$$y = 2\frac{x-a}{b-a}-1$$
• Orthogonality holds with weights function with the same change of variable
#### Chebyshev approximation algorithm¶
1. Given $f(x)$ and $[a,b]$
2. Compute Chebyshev interpolation nodes on $[-1,1]$
3. Adjust nodes to $[a,b]$ by change of variable, $x_i$
4. Evaluate $f$ at the nodes, $f(x_i)$
5. Compute Chebyshev coefficients $a_i = g\big(f(x_i)\big)$
6. Arrive at approximation
$$f(x) = \sum_{i=0}^n a_i T_i(x)$$
In [16]:
import numpy.polynomial.chebyshev as cheb
for degree, clr in (7,'b'),(9,'g'),(11,'m'):
fi=cheb.Chebyshev.interpolate(func,degree,[-5,10])
plot1(fi,fdata=(None,None),color=clr,label='Chebyshev with n=%d'%degree,extrapolation=True)
### Multidimensional interpolation¶
• there are multidimensional generalization to all the methods
• curse of dimensionality in the number of interpolation points when number of dimensions increase
• sparse Smolyak grids and adaptive sparse grids
• irregular grids require computationally expensive triangulation in the general case
• good application for machine learning!
Generally much harder! | {
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## Decimal To Binary | {
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In order to convert a decimal number to its binary equivalent, we will repeatedly divide the decimal number by 2, the base of the binary system. One for "yes", and zero for "no". Binary definition, consisting of, indicating, or involving two. Add the following byte-long (8 bit) two’s complement numbers together, and then convert all binary quantities into decimal form to verify the accuracy of the addition:. This tutorial explains how to convert a decimal IP address in binary IP address and a binary IP address in a decimal IP address step by step with examples. A value can also be in binary, hexadecimal or octal if the appropriate prefix is included. The number that i'm trying to convert is a source address in a decimal form. How to Convert Decimal to Binary and vise versa is crucial to master, such skills will give you the confidence when you deal with Network and Storage devices. It allows the user to specify the bit of the binary number that will be converted to. Example of decimal to binary conversion: The decimal positive integer 330 can be deconstructed: 330 10 = 3*100 + 3*10 + 0*1 = 1*10 2 + 2*10 1 + 5*1 0. Before a floating-point binary number can be stored correctly, its mantissa must be normalized. Converting between Decimal (Base 10) and Binary (Base 2) Base 10 vs. In this tutorial i am going to show you how to convert “Decimal to Binary”, “Decimal to Octal” and “Decimal to Hexadecimal” Check the Java Program below. For example, the decimal. In the octal to binary conversion method, we will convert each digit from the given octal number into three bit binary number starting from the right most octal digit(LSD) to the left most digit(MSD) and at the end combine each three bit binary number to form the binary equivalent of given octal number. Program to convert decimal to binary more than 18 bits. Example :- 7 1). This way people won't think it is the decimal number "101" (one hundred and one). Below is the code for a 3-digit BCD-Counter (binary Coded | {
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number "101" (one hundred and one). Below is the code for a 3-digit BCD-Counter (binary Coded decimal) I implemented while studying verilog. C program to covert 'Decimal to Binary' and 'Binary to Decimal' - Binary Decimal Conversion. Video created by Hebrew University of Jerusalem for the course "Build a Modern Computer from First Principles: From Nand to Tetris (Project-Centered Course)". This is how we would successfully convert a binary number into decimal, (which you can read about here) but we actually want to do the reverse, and easily convert decimal to binary. Binary to Decimal to BCD Conversion Example. Explanation of how number bases, including binary and decimal, are used for reasoning about digital data. Take this quiz to test yourself on conversion between decimal and binary representations. This way you can convert up to 19 decimal characters (max. To convert fraction to binary, start with the fraction in question and multiply it by 2 keeping notice of the resulting integer and fractional part. Decimal -> Binary Decimal to binary conversion is even more exciting. You can convert to and from binary and the base-10 system typically used by humans. The binary number 1101011001111110 converts to 54910 in Decimal and D67E in hex. Converting decimal fraction to binary. In this example I use 8 bit binary numbers, but the principle is the same for both 8 bit binary numbers (chars) and 32 bit binary numbers (ints). Binary, and other systems of representing numbers is similar to decimal, only the use a different amount of symbols or digits to represent a number. It is much easier to work with large numbers using hexadecimal values than decimal. binary values The following table shows the decimal values of binary numbers. Program to Convert Binary to Decimal number:Number System [crayon-5d4e448ac93bb860279627/] Output : [crayon-5d4e448ac93c5762542184/] Note : This program is for beginners (not for Experts) that’s why we haven’t provided any Validations while | {
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This program is for beginners (not for Experts) that’s why we haven’t provided any Validations while accepting the input. Type 110 in binary by clicking the Buttons "1", "1" and "0" in the binary system calculator. 243F6 in a hexadecimal system (base 16), or to 11. It can convert very large and very small numbers — up to hundreds of digits. Hexadecimal is used to convert byte/modern computer numbers into defined binary digits. We use cookies for various purposes including analytics. It provides output as 170 Decimal value. For beginners hexadecimal is base 16 number, while decimal is base 10, Octal is base 8 and binary is base 2 numbers in Number systems. Method1: Use (()) brace expatiation. This video tutorial explains how to convert decimal to binary numbers. It focuses on the following learning objectives: 5b. As it is written, it would be hard to work out whether it is a binary or decimal (denary) value. Each hexadecimal digit represents four binary digits (bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a human-friendly representation of binary coded values in computing and digital. Convert any decimal number between 0 and 255 into binary. Decimal to Binary, Hex & Octal Converter to perform decimal to binary, decimal to hex & decimal to octal conversion online by using simple successive division methods along with step by step calculation & solved example problems. The online BCD to Binary Converter is used to convert BCD (Binary-coded decimal) to binary (Base-2) number. AD3, Alternating Directions Dual Decomposition. It involves dividing the number to be converted, say N, by 2 (since binary is in base 2), and making note of. Then the calculate button is pressed and the textbox should display the corresponding decimal number. TCS Programming Question Solution - 1. Two good examples where we use hexadecimal values are MAC addresses and IPv6 addresses. So the binary (so far) is _ _ _ 1 1. Decimal To Binary Conversion A decimal | {
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and IPv6 addresses. So the binary (so far) is _ _ _ 1 1. Decimal To Binary Conversion A decimal number system is a base 10 number system. Likewise, we use 0 and 1 to write numbers in the binary number system. Calculator will display the result in total number of hours, minutes and seconds in decimal. This guide shows you how to convert from decimal to binary and binary to decimal. Enter the primary number (in binary; make sure it is valid) first then enter the secondary number (also in binary) for the calculation and click on Calculate. Binary to decimal - Four ways to convert formats in Matlab In this article we're going to present four variations of a binary to decimal conversion in Matlab; we're going to convert binary numbers (numbers with only symbols '0' and '1') to decimal numbers (numbers with 10 different symbols, from '0' to '9'). Binary to hexa decimal decoder / converter. 0 Flash Drive Cute Memory Stick Fruit Vegetable Series Thumb Drive Data Storage Pendrive Cartoon Jump Drive Gift: USB Flash Drives - Amazon. In this system, numbers are represented in a decimal form, however each decimal digit is encoded using a four bit binary number. the radix of binary number system). Specifically, dotted decimal notation is used in various IT contexts, including in Internet Protocol addresses. For example-. Roman to decimal converter. Say we wish to convert an unsigned decimal integer, $$N\text{,}$$ to binary. Binary form of 15 is 1111 Binary form of 10 is 1010 Binary form of 18 is 10010 Binary form of 27 is 11011 In the above program, the DecimalToBinary function has binary value of the decimal number n and is stored in the array binaryNumber[]. Although computers are based on the binary number system, we don’t have to use binary numbers when using one. Especially for IPv6 addresses it's useful to understand how you can calculate from hexadecimal to binary and decimal or the other way around. We use decimal number system for our day to day calculations. C Program | {
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or the other way around. We use decimal number system for our day to day calculations. C Program to Convert Decimal to Binary. C++ program to convert decimal number into binary. However there are alternate solutions, in case you were just curious to know :) Method #2 [code. Decimal to Binary Table. Decimal -> Binary Decimal to binary conversion is even more exciting. Binary string to decimal conversion. It will convert a decimal number to its nearest single-precision and double-precision IEEE 754 binary floating-point number, using round-half-to-even rounding (the default IEEE rounding mode). The equivalent decimal representation of a binary number is the sum of the powers of 2 which each digit represents. Binary number is a base 2 number because it is either 0 or 1. Binary Fractions Summary. However, if you study the Microsoft Windows calculator application that ships along with the OS, the "programmer" type of calculator takes an input whether the entered number is binary or decimal from the user, then internally in its code it would check whether the entered number was a valid binary or decimal number. Example values of hexadecimal numbers converted into binary, octal and decimal. These 8-bits can represent any decimal number between 0 and 255. The decimal number is equal to the sum of binary digits (d n) times their power of 2 (2 n):. There are no ads, popups or nonsense, just an awesome decimal number to BCD number converter. IP addressing is a core functionality of networking today. Take a look at decimal number 6. And I encourage you to pause the video, and try to work through it out on your own. Press button, get result. Decimal : For denoting integer and non-integer numbers, decimal number system uses 10 different digits, 0,1,2,3,4,5,6,7,8 and 9 i. Number conversion between hexadecimal and decimal. How to Convert from Decimal to Binary. C Program To Convert Decimal To Octal Number. If you work with computers, you may find yourself needing a basic | {
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Convert Decimal To Octal Number. If you work with computers, you may find yourself needing a basic understanding of binary. Entering Repeating Decimals. Binary numbers can contain digits 0 and 1 but no decimal or scientific notation. Here we will learn, How to convert decimal to binary in java with easy examples. Let me show it on example. In binary form, for example, the decimal quantity 1895 appears as 11101100111. If you’d like to dig deeper into Binary, this course helps explain the details, and show how Binary is the foundation of all digital computing. dec = Fix(dec) / 2: After obtaining the first residue the variable "dec" is going to get itself value between 2. Binary Coded Decimal which is also called as BCD is another process for converting decimal numbers into their binary equivalents. Binary numbers have lot of use in the digital world, in fact, binary is the language of computers where 0 and 1 represent true/false, on/off and becomes key for logic formation. In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 2 0, the next representing 2 1, then 2 2, and so on. Continue multiplying by 2 until you get a resulting fractional part equal to zero. Treat the division as an integer division. binary values The following table shows the decimal values of binary numbers. Then just add them up. It then displays the decimal values and the hexadecimal value of the elements in the array returned by the GetBits method. 0x578FCF (hex) 5738447 (decimal) 025707717 (octal) 10101111000111111001111 (binary) are all different representations of the same number. You can use national decimal items anywhere that other category numeric data items can be used. 25), 2-3 (0. By using user defined logic. About the Decimal/Binary Converter. The binary or base 2 numbering system is the keystone of computer systems and digital electronics. Input and output in binary, decimal, hexadecimal or ASCII. Get the integer quotient for the | {
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Input and output in binary, decimal, hexadecimal or ASCII. Get the integer quotient for the next iteration. The following function takes in a binary string and returns a bigint with the decimal value:. Decimal to Binary Using Excel: https://www. To use this decimal to binary converter tool, you should type a decimal value like 308 into the left field below, and then hit the Convert button. Write a C program to input any decimal number from user and convert it to binary number system using bitwise operator. A one has the value 1 and a zero has the value 0. This shweet conversion tool will take any text string and convert it into binary code - you know? those little 1's and 0's that make our world go around. 2) Convert the decimal number 143 into binary. Hello guys, Can anyone give me the code required for 'decimal to binary conversion using verilog for 4 bit' please im doing a project. 14159 in a decimal system (base 10), or to 3. Convert number systems units. Department of Labor’s Employment and Training Administration. Watch our tutorial on How To Do Decimals To Binary Numbers from one of Videojug's professional experts. (Binary to decimal). ; In this method of conversion the fractional part of the given decimal number is multiplied by 2 (i. Binary number: In digital electronics and mathematics, a binary number is a number expressed in the base-2 numeral system or binary numeral system. The Quotient method is the new conversion method invented and derived by using Raghavendra's Analysis. There are other methods also available which can also be used in decimal to binary conversion. Convert Binary to Decimal Numbers. Let's see if we can get some experience converting from a decimal representation to a binary representation. Binary form base 8 Octal form base 10 Decimal form (common used) base 16 Hexadecimal form (hexa) Examples. This is a conversion table with decimal numbers next to their binary and hex equivalents. It means that a binary number can be formed by | {
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numbers next to their binary and hex equivalents. It means that a binary number can be formed by using 0 and 1 (2 base elements) only. Binary to Decimal conversion How to convert decimal to binary Conversion steps: Divide the number by 2. Useful, free online tool that converts plain text to decimal string. Fast Decimal to Binary Conversion. The flip side of the coin is translating a decimal number into binary format. (The old flash version is here. What that means is, everytime you count numbers under base 10, you are counting from zero to nine, then starting over by adding another number in front to make 10 and so on. It casts a string following it to a binary string. Several numeric data types are discussed, including the common "packed" and "comp-3" fields. The computer chips that run PCs work with a binary system. Binary arithmetic is pretty easy once you know what's going on. A circuit is either on or off. 33) convert to binary Ask for details ; Follow Report by Arjun306 7 minutes ago Log in to add a comment. The representation of such a number in decimal and binary and any other such system must be infinite in length and non-periodic. Converting Decimal Fractions to Binary. 625) a_split = (int(a//1),a%1) Convert the fractional part in its binary representation. Program to convert decimal to binary more than 18 bits. A number only becomes hexadecimal, decimal or binary when you format it in a certain way for printing. The right-most. s, To binary is easy: each digit after the decimal point (to the left) represents powers of 2. In the example a 1, 0, and then 1 is added to the listbox. Learn how to convert 26 to binary and create a Java project to convert from base-10 to base-2. decimal synonyms, decimal pronunciation, decimal translation, English dictionary definition of decimal. It takes three binary digits to represent an octal digit. > convert_to_binary(52) 110100 In this program, we convert decimal number entered by the user into binary using a recursive | {
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110100 In this program, we convert decimal number entered by the user into binary using a recursive function. 3424 (base 10) = 110101100000 (base 2). This way people won't think it is the decimal number "101" (one hundred and one). I understand the arithmetics of it, but I was wondering how I can add the remainders onto each other using strings. For example, say that N is 23. Here you can make an arithmetic calculation you are interested in values through our online calculators. An understanding of binary numbers,the binary system, and how to convert between binary and decimal is essential for anyone involved in computers, coding, and networking. Below is the code for a 3-digit BCD-Counter (binary Coded decimal) I implemented while studying verilog. For example, decimal 1234. Created by computer nerds from team Browserling. Binary to Decimal Conversion To convert binary number into decimal number we need to multiply each bit of binary by increasing powers of 2 starting from 0. We will use the bitwise operator "AND" to perform the desired task. Just a quick recap, binary is a base-2 number system that uses 0 and 1 as the only digits, and hexadecimal is a base-16 number system that uses sixteen values from 0 to 9 and a to f as digits. , a number with fractional part. In a base 10 (or decimal) number, each column in the number stands for a power of 10. Decimal to binary conversion: two methods to do it with Matlab In this article we're going to present two methods of a decimal to binary conversion in Matlab; that is, we're going to convert decimal numbers (numbers with 10 different symbols, from '0' to '9', or in base 10) into binary numbers (numbers with only symbols '0' and '1', or in base 2). Converting Decimal Fractions to Binary. Binary numbers can contain digits 0 and 1 but no decimal or scientific notation. Each hexadecimal digit represents four binary digits (bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a human-friendly | {
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(bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a human-friendly representation of binary coded values in computing and digital. DECIMAL TO BINARY / HEXADECIMAL TO BINARY synonyms, DECIMAL TO BINARY / HEXADECIMAL TO BINARY pronunciation, DECIMAL TO BINARY / HEXADECIMAL TO BINARY translation, English dictionary definition of DECIMAL TO BINARY / HEXADECIMAL TO BINARY. How to Show that a Number is Binary. Repeat step 2 and 3 until result is 0. This way people won't think it is the decimal number "101" (one hundred and one). Java program to convert decimal to binary. The negative of this number is the value of the original binary. The decimal number is equal to the sum of binary digits (d n) times their power of 2 (2 n):. Converting a binary floating point number to decimal. So for example, 10111 is actually equal to 2^4+2^2+2^1+2^0 (2^3 is not included because there is a zero in that spot). It involves dividing the number to be converted, say N, by 2 (since binary is in base 2), and making note of. Welcome to the website Decimal-to-Binary. Example values of hexadecimal numbers converted into binary, octal and decimal. 2) Do conversion by writing your own logic without using any predefined methods. I'm having trouble writing the program, specifically finding a function that converts the items in the listbox to a decimal number. We will see two Python programs, first program does the conversion using a user defined function and in the second program we are using a in-built function bin() for the decimal to binary conversion. Binary has a base of 2. I've been trying to convert the number -441 to binary, but I don't really know how I can accomplish this. This wikiHow will show you how to do this. Returns a string representation of an integer or of a binary type converted to hexadecimal. To show that a number is a binary number, follow it with a little 2 like this: 101 2. How to Convert Decimal to Binary and vise versa is crucial to | {
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it with a little 2 like this: 101 2. How to Convert Decimal to Binary and vise versa is crucial to master, such skills will give you the confidence when you deal with Network and Storage devices. Decimal to Base 2 Algorithm. Input and output in binary, decimal, hexadecimal or ASCII. Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two. Byte conversion chart for binary and decimal conversion In Data storage and when describing memory size, a Kilobyte is 2^10, or 1024 bytes. Also Read: Convert Binary to Decimal in Java. Java Convert Decimal to Binary example and examples of string to int, int to string, string to date, date to string, string to long, long to string, string to char, char to string, int to long, long to int etc. This guide shows you how to convert from decimal to binary and binary to decimal. There are many number conversion interview questions which is asked in java interviews like java program to convert decimal to hexadecimal, java program to convert decimal to. Python Program to Check Whether a String is Palindrome or Not; Python Program to Multiply Two Matrices; Python Program to Convert Decimal to Binary Using Recursion; Python Program to Find Factorial of Number Using Recursion; Python Program to Convert Decimal to Binary, Octal and Hexadecimalc; Python Program To Display Powers of 2 Using. A binary number which consists of only 1 and 0, while a decimal number is a number which consists of numbers between 0-9. How computers see IP addresses. 1 mm to decimal = 1 decimal. Quite some time ago I had a particular need to convert Decimal to Binary and vice versa. Repeat this process till quotient becomes zero. Keep calling conversion function with n/2 till n > 1, later perform n % 1 to get MSB of converted binary number. hello i have a string with 11 bits in it in binary and would like to make this into a decimal how can i do it the stuff in the string | {
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in it in binary and would like to make this into a decimal how can i do it the stuff in the string looks like this 10111100111 and when this is converted to a decimal should be 3023. 30 mm to decimal = 30 decimal. Converts decimal -100 to binary (1110011100) Convert a decimal number to hexadecimal. Java program, asking user for a positive integer and converting it to binary representation. ) Instructions Just type in any box, and the conversion is done "live". Steps for using the binary decoder tool: First select the type of data you want the binary code to be converted. sage code for decimal to binary expansion. In hexadecimal, each digit can be 16 values, not 10. Decimal Code Binary; Decimal Coded Wire. Two’s complement notation really shows its value in binary addition, where positive and negative quantities may be handled with equal ease. Calculator for crc32, md5, sha1, ripemd128 and gost hash algorithms. Binary is the numeral system used to express data stored in computers. Practice these questions for upcoming IBPS PO Mains Examination. Then the calculate button is pressed and the textbox should display the corresponding decimal number. Converting binary to decimal (base-2 to base-10) or decimal to binary numbers (base10 to base-2) can be done in a number of different ways as shown above. Scanner; public class Decimal_Binary { Scanner scan; int num; void getVal() { System. About the Decimal/Binary Converter. Hi, I was asked to program a recursion (in C), which converts a binary to decimal. I've been trying to convert the number -441 to binary, but I don't really know how I can accomplish this. Hence, negative binary numbers are used to represent On or Off in electronic circuits. The maximum value of a 8 bit binary number is 255 in decimal. Unit Descriptions; 1 Gigabyte (binary): A gigabyte (binary) contains 1024 3 bytes, this is the same as a gibibyte. Example values of hexadecimal numbers converted into binary, octal and decimal. You can write it like this: | {
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values of hexadecimal numbers converted into binary, octal and decimal. You can write it like this: Yes, I've made it up. Watch our tutorial on How To Do Decimals To Binary Numbers from one of Videojug's professional experts. Then the calculate button is pressed and the textbox should display the corresponding decimal number. 3020 3100 3370 3610 Question 4: Convert 11101111 from binary to decimal. Instead of using a base of 10 or 2 respectively, it uses a base of 16. Of course, a 32-bit binary cannot hold the full range of values that you can get from 12 decimal digits. Now press " x " button from the common operators box. program to convert decimal number to binary DESCRIPTION-: This program receive input as decimal number and then calculate its binary equivalent and then display the binary equivalent of that decimal number. - [Voiceover] Let's now see if we can convert a larger decimal representation to binary. To indicate our answer is in binary (binary numbers are base 2), we must include a 2 at the bottom right corner of the sequence. Binary number system is the basis of all computer and digital systems. As written, IniVal is declared, but only temp is declared and typed as string. Hexadecimal number system: It is base 16 number system which uses the digits from 0 to 9 and A, B, C, D, E. Binary, octal and hexadecimal number systems are closely related and we may require to convert decimal into these systems. If it's odd, write a 1. Please feel free to edit/adapt this before posting. Convert time hh:mm:ss to decimal hours, decimal minutes and total seconds. Instead, we enter decimal numbers the computer converts into binary before manipulating them. An Algorithm for Converting a Decimal Number to a Binary Number by Fox Valley Technical College is licensed under a Creative Commons Attribution 4. CBinS16 – converts a decimal signed integer to 16 bit binary string. The numbering system everyone is most familiar with is base 10, also known as decimal or denary. This | {
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numbering system everyone is most familiar with is base 10, also known as decimal or denary. This is a discussion of COBOL Computational fields. First you need to define the properties of the 16 bit binary encoding that you want to use. Calculating binary numbers can be confusing, until you figure out the system. Java program to convert decimal to binary. Decimal to hexadecimal conversion method: Following steps describe how to convert decimal to hexadecimal Step 1: Divide the original decimal number by 16 Step 2: Divide the quotient by 16 Step 3: Repeat the step 2 until we get quotient equal to zero. Hi i am writing a program to convert a user entered integer number into its binary representation, the aim is to repeatedly ask the user to enter a value then output the number in binary and when 9999 is entered end program, i've managed to make it work with positive integers so far only it displays. Compiled on Platform: Windows XP Pro SP2. 001 is decimal. Decimals are nothing more than glorified fractions. If you’d like to dig deeper into Binary, this course helps explain the details, and show how Binary is the foundation of all digital computing. Can you describe a method of converting a binary number to decimal? 2. June 24, 2019. Following fig-1 mentions block diagram of decimal to binary labview vi. 0 to 255) Convert any binary number from 0 to 111111111 into octal; Convert any octal number from 0 to 777 into binary; Convert any binary number from 0 to 1111 1111 1111 1111 into hex. Using Command Line Arguments Program to Convert a Decimal to Binary Number. Hexadecimal is used to convert byte/modern computer numbers into defined binary digits. The program for conversion of decimal to binary depends on the problem specification. Create a program that would convert a decimal number to binary, octal or hexadecimal counterpart. Write your instructions in English. Converting Decimal to Binary numbers can be done in a number of different ways as shown above. 121 163 199 | {
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Decimal to Binary numbers can be done in a number of different ways as shown above. 121 163 199 212 Question 2: Convert 101101 from binary to decimal. Let's see if we can convert the number 13 in decimal to binary. Decimal to Binary, Hex & Octal Converter to perform decimal to binary, decimal to hex & decimal to octal conversion online by using simple successive division methods along with step by step calculation & solved example problems. In this system, numbers are represented in a decimal form, however each decimal digit is encoded using a four bit binary number. All the best and keep practicing!. Before a floating-point binary number can be stored correctly, its mantissa must be normalized. We set it equal to the expression in Equation (2. A small point, but IniVal is NOT typed as a string variable in your code. Given a decimal number as input, we need to write a program to convert the given decimal number into equivalent binary number. Example for Binary to Decimal Conversion. In this post, we will see programs to convert decimal number to an equivalent binary number. Decimal ---> Base 10. Repeat the steps until the quotient is equal to 0. Step 2: Divide 10 by 2. decimal synonyms, decimal pronunciation, decimal translation, English dictionary definition of decimal. 500 * 10 9 bytes = 500,000,000,000 = 500 Gigabytes But the operating system determines the size of the drive using the computer's binary powers of two definition of the "Giga" prefix:. Like I said a hexadecimal number doesn't actually name a type: a type is something like long, or int. Hexadecimal is base 16. I first converted 441 to binary which is: 110111001 Then I'm supposed to take the complim. If everything is done right, the result should be 34. Java decimal to binary conversion is the most important core java interview question. 7/2 = Quotient = 3(grater than 1), Remainder = 1. Conversion Hexa to Decimal. Source code DecimalToBinaryConverter. The hexadecimal number system (hex) functions | {
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Hexa to Decimal. Source code DecimalToBinaryConverter. The hexadecimal number system (hex) functions virtually identically to the decimal and binary systems. In computing and electronic systems, binary-coded decimal (BCD) is a digital encoding method for decimal numbers in which each digit is represented by its own binary sequence. You rely on decimal numbers when doing calculations for your business, such as when figuring sales tax or creating a monthly budget. How to Convert Decimal to Binary and vise versa is crucial to master, such skills will give you the confidence when you deal with Network and Storage devices. Convert 13 10 to binary:. If you understand binary because you learned it for understanding how IP addresses work, you usually stop at 2 7 because one octet of an IP address is 8 bits long. Say we wish to convert an unsigned decimal integer, $$N\text{,}$$ to binary. The binary system is the internal language of electronic computers. Set s=0 for a positive number and s=1 for a negative number. How to convert binary to decimal: The binary number system, also known as the base 2 number system; is used by all modern generation computers internally. Online IEEE 754 floating point converter and analysis. BCD-To-Decimal Decoder Binary-To-Octal Decoder The MC14028B decoder is constructed so that an 8421 BCD code on the four inputs provides a decimal (one−of−ten) decoded output, while a 3−bit binary input provides a decoded octal (one−of−eight) code output with D forced to a logic “0”. This free binary calculator can add, subtract, multiply, and divide binary values, as well as convert between binary and decimal values. It took a little digging but eventually came across this little gem and thought it was worth posting for others to use. For more information about the binary representation of Decimal values and an example, see the Decimal(Int32[]) constructor and the GetBits method. How to convert from decimal number system to binary number system using bitwise | {
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the GetBits method. How to convert from decimal number system to binary number system using bitwise operator in C programming. Decimal to binary converter Abstract. Decimal : For denoting integer and non-integer numbers, decimal number system uses 10 different digits, 0,1,2,3,4,5,6,7,8 and 9 i. The binary number is constructed from right to left. Convert from binary to decimal algorithm: For this we multiply each digit separately from right side by 1, 2, 4, 8, 16 … respectively then add them. Decimal conversions can be done by either successive division method or successive multiplication method. Then enter or paste your binary code in the first text box and click Decode button. (The old flash version is here. And I encourage you to pause the video, and try to work through it out on your own. The following utility converts the IP (TCP/IP) address to other browser URL addressable forms. Hexadecimal --> Base 16. For more information about the binary representation of Decimal values and an example, see the Decimal(Int32[]) constructor and the GetBits method. This page of labview source code covers decimal to binary labview vi which converts decimal vector to binary vector. 0,1,2,3,4,5,6,7,8,9 are decimal number and all other numbers are based on these 10 numbers. Here you can make an arithmetic calculation you are interested in values through our online calculators. Binary --> Base 2. The syntax is similar to that of the ToInt32 method. , a number with fractional part. That is the most confusing thing that I want to say about the subject. Jun 9, It was a fascinating rabbit hole that exposed me to things like “decimal machines” and “2 out of 5 code”. Short for binary-coded decimal, BCD is also known as packet decimal and is numbers 0 through 9 converted to four-digit binary. So Java Code Online is going to discuss a case when a number entered as binary is to be converted to a decimal number. Dotted decimal notation is a system of presenting numbers that is a little | {
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to a decimal number. Dotted decimal notation is a system of presenting numbers that is a little different from the common conventions in arithmetic as it is taught in schools. Here you can find the answer to questions like: Convert decimal number 168 in binary or Decimal to binary conversion. The Land Before Binary. | {
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# Maximum Prefix Mex Sum - Editorial
Setter(s) Divyansh Verma
Tester(s) Vichitr Gandas, Mayank Pugalia
Difficulty Easy
Topics Observation, Greedy, Maths, Combinatorics
## Solution Idea
The prefix Mex sum is maximum when we put all unique numbers in increasing order at the start. Once a number is missed from 0,1,2..., this chain breaks and then Mex will remain same for rest of the prefix arrays. Example:
A=[0,1,2,5,7]
Here Mex for first three prefix arrays is 1,2,3 respectively. After that we see that 3 is not present. Hence for rest of subarrays Mex will remain 3 only.
This way, the sum is maximized when we arrange initial unique numbers in order 0,1,2... like this. And if this order breaks or we are left with duplicate items, we can permute them in any way. Here in the beginning, each unique number x can be picked with count[x] ways where count[x] is number of times it occurs in the array. And let’s say number of remaining elements after break is y, they can be arranged in y! ways. Hence total number of permutations with max prefix Mex sum would be count[0]*count[1]*...*count[k]*y! where k is first number where this sequence 0,1,2,... breaks.
Count can be maintained with map. We can reduce the time complexity here by using a vector instead because the sequence if continued won’t exceed n and frequency of rest of the items is not needed, because they will be part of left over items which can be permuted.
### Complexity Analysis
Time Complexity: \mathcal{O}(n \log{n}) for pre calculation and to find the answer which can be reduced to \mathcal{O}(n) using a vector instead of map.
Space Complexity: \mathcal{O}(n) for maintaining the frequency.
## Codes
Setter's Code
#include<bits/stdc++.h>
using namespace std;
#define int long long int
const int MAXN = 1e5 + 10;
const int MOD = 1e9 + 7;
int fact[MAXN];
void preCompute(){
fact[0] = 1;
for(int i=1;i<MAXN;i++) fact[i] = (fact[i-1]*i)%MOD;
}
class FindMexMax {
public:
int getOutput(vector<int> a){ | {
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class FindMexMax {
public:
int getOutput(vector<int> a){
map<int,int> which;
for(int v:a) which[v]++;
int cnt = a.size(),ans = 1;
for(int i=0;i<a.size();i++){
if(which[i] == 0) break;
ans = (ans*which[i])%MOD;
which[i]--;
cnt--;
}
ans = ans*fact[cnt]%MOD;
return ans;
}
};
signed main(){
preCompute();
int t;
cin>>t;
while(t--){
int n; cin>>n;
vector<int> a(n);
for(int &v:a) cin>>v;
FindMexMax fMM;
cout<<fMM.getOutput(a)<<endl;
}
}
Tester's Code
/***************************************************
@author: vichitr
Compiled On: 13th Mar 2021
*****************************************************/
#include<bits/stdc++.h>
#define endl "\n"
#define int long long
using namespace std;
const int MOD = 1e9 + 7;
const int N = 1e5 + 7;
int fact[N];
void init(){
fact[0] = 1;
for(int i=1;i<N;i++){
fact[i] = (fact[i-1] * i) % MOD;
}
}
void solve(){
int n; cin>>n;
assert(n <= 1e5);
int a[n];
map<int, int> M;
for(int i=0;i<n;i++){
cin>>a[i];
assert(a[i] >= 0 and a[i] <= 1e9);
M[a[i]]++;
}
int ans = 1;
int tot = 0;
for(int i = 0; i < n; i++){
if(M[i] == 0){
break;
}
ans *= M[i];
ans %= MOD;
tot++;
}
ans *= fact[n - tot];
ans %= MOD;
cout<<ans<<'\n';
}
signed main(){
int t = 1;
cin>>t;
assert(t <= 5);
init();
while(t--){
solve();
}
return 0;
}
If you have used any other approach, share your approach in comments! | {
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# True or False. Every Diagonalizable Matrix is Invertible
## Problem 439
Is every diagonalizable matrix invertible?
## Solution.
### Counterexample
We give a counterexample. Consider the $2\times 2$ zero matrix.
The zero matrix is a diagonal matrix, and thus it is diagonalizable.
However, the zero matrix is not invertible as its determinant is zero.
### More Theoretical Explanation
Let us give a more theoretical explanation.
If an $n\times n$ matrix $A$ is diagonalizable, then there exists an invertible matrix $P$ such that
$P^{-1}AP=\begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix},$ where $\lambda_1, \dots, \lambda_n$ are eigenvalues of $A$.
Then we consider the determinants of the matrices of both sides.
The determinant of the left hand side is
\begin{align*}
\det(P^{-1}AP)=\det(P)^{-1}\det(A)\det(P)=\det(A).
\end{align*}
On the other hand, the determinant of the right hand side is the product
$\lambda_1\lambda_2\cdots \lambda_n$ since the right matrix is diagonal.
Hence we obtain
$\det(A)=\lambda_1\lambda_2\cdots \lambda_n.$ (Note that it is always true that the determinant of a matrix is the product of its eigenvalues regardless diagonalizability.
See the post “Determinant/trace and eigenvalues of a matrix“.)
Hence if one of the eigenvalues of $A$ is zero, then the determinant of $A$ is zero, and hence $A$ is not invertible.
The true statement is:
a diagonal matrix is invertible if and only if its eigenvalues are nonzero.
### Is Every Invertible Matrix Diagonalizable?
Note that it is not true that every invertible matrix is diagonalizable. | {
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Note that it is not true that every invertible matrix is diagonalizable.
For example, consider the matrix
$A=\begin{bmatrix} 1 & 1\\ 0& 1 \end{bmatrix}.$ The determinant of $A$ is $1$, hence $A$ is invertible.
The characteristic polynomial of $A$ is
\begin{align*}
p(t)=\det(A-tI)=\begin{vmatrix}
1-t & 1\\
0& 1-t
\end{vmatrix}=(1-t)^2.
\end{align*}
Thus, the eigenvalue of $A$ is $1$ with algebraic multiplicity $2$.
We have
$A-I=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ and thus eigenvectors corresponding to the eigenvalue $1$ are
$a\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ for any nonzero scalar $a$.
Thus, the geometric multiplicity of the eigenvalue $1$ is $1$.
Since the geometric multiplicity is strictly less than the algebraic multiplicity, the matrix $A$ is defective and not diagonalizable.
### Is There a Matrix that is Not Diagonalizable and Not Invertible?
Finally, note that there is a matrix which is not diagonalizable and not invertible.
For example, the matrix $\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ is such a matrix.
## Summary
There are all possibilities.
1. Diagonalizable, but not invertible.
Example: $\begin{bmatrix} 0 & 0\\ 0& 0 \end{bmatrix}.$
2. Invertible, but not diagonalizable.
Example: $\begin{bmatrix} 1 & 1\\ 0& 1 \end{bmatrix}$
3. Not diagonalizable and Not invertible.
Example: $\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}.$
4. Diagonalizable and invertible
Example: $\begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix}.$
Determine whether each of the following statements is True or False. (a) If $A$ and $B$ are $n \times n$... | {
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# $\mathbb{Z}_2\times\mathbb{Z}_2$ has order $4$? How so?
$\mathbb{Z}_2\times\mathbb{Z}_2$ has order $4$, the neutral element is $(0_2,0_2)$ and the other elements have order $2$. Therefore, $\mathbb{Z}_2\times\mathbb{Z}_2$ is not cyclic,so it is the Klein group.
The elements of $\mathbb{Z_2}$ are $\{0_2,1_2\}$ and the order of $1_2$ is 2.
If we take the direct product $\mathbb{Z}_2\times\mathbb{Z}_2$ then we have a generator of the Group product $\langle 1_2,1_2\rangle$.
Question:
How can the author state $\mathbb{Z}_2\times\mathbb{Z}_2$ has order $4$? Is $\langle 1_2,1_2\rangle$ not a generator of order 2 of $\mathbb{Z}_2\times\mathbb{Z}_2$?
• Remember that the order of a group is a somewhat distinct concept from the order of an element. The order of a group is the cardinality of that group (in this case, 4---the elements can be explicitly written fairly easily), while the order of an element $a$ is the smallest natural number $n$ such that $a^n$ is the identity (here, the maximal order of an element is 2). – Xander Henderson Nov 29 '17 at 18:54
• The order of a group means its size. – Ittay Weiss Nov 29 '17 at 18:54
• $(1,1)$ does not generate the element $(0,1)$ – Doug M Nov 29 '17 at 18:55
• $\langle 1, 1\rangle$ only generates all of $\Bbb Z_n\times \Bbb Z_m$ if the greatest common factor of $n$ and $m$ is 1. Here it is 2. – MJD Nov 29 '17 at 18:57
The order of a group is the number of elements in it. $\mathbb{Z}_2\times\mathbb{Z}_2$ has four elements: $(0,0), (0,1), (1,0), (1,1)$.
But as you have seen, each element has order $\le 2$. (This, in particular, proves that $\mathbb{Z}_2\times\mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$.)
The elements of $\mathbb{Z}_2 \times \mathbb{Z}_2$ are | {
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# Difference between revisions of "1995 AHSME Problems/Problem 27"
## Problem
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
$$\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular}$$
Let $f(n)$ denote the sum of the numbers in row $n$. What is the remainder when $f(100)$ is divided by 100?
$\mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 }$
## Solution 1
Note that if we re-draw the table with an additional diagonal row on each side, the table is actually just two of Pascal's Triangles, except translated and summed. $$\begin{tabular}{ccccccccccccccc} & & & & & 1 & & 0 & & 1 & & & & \\ & & & & 1 & & 1 & & 1 & & 1 & & & \\ & & & 1 & & 2 & & 2 & & 2 & & 1 & & \\ & & 1 & & 3 & & 4 & & 4 & & 3 & & 1 & \\ & 1 & & 4 & & 7 & & 8 & & 7 & & 4 & & 1 \\ 1 & & 5 & & 11 & & 15 & & 15 & & 11 & & 5 & & 1 \end{tabular}$$
The sum of a row of Pascal's triangle is $2^{n-1}$; the sum of two of each of these rows, subtracting away the $2$ ones we included, yields $f(n) = 2^n - 2$. Now, $f(100) = 2^{100} - 2 \equiv 2 \pmod{4}$ and $f(100) = 2^{100} - 2 \equiv 2^{20 \cdot 5} - 2 \equiv -1 \pmod{25}$, and by the Chinese Remainder Theorem, we have $f(100) \equiv 74 \pmod{100} \Longrightarrow \mathrm{(E)}$.
## Solution 2 (induction)
We sum the first few rows: $0, 2, 6, 14, 30, 62$. They are each two less than a power of $2$, so we try to prove it: | {
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Let the sum of row $n$ be $S_n$. To generate the next row, we add consecutive numbers. So we double the row, subtract twice the end numbers, then add twice the end numbers and add two. That makes $S_{n+1}=2S_n-2(n-1)+2(n-1)+2=2S_n+2$. If $S_n$ is two less than a power of 2, then it is in the form $2^x-2$. $S_{n+1}=2^{x+1}-4+2=2^{x+1}-2$.
Since the first row is two less than a power of 2, all the rest are. Since the sum of the elements of row 1 is $2^1-2$, the sum of the numbers in row $n$ is $2^n-2$. Thus, using Modular arithmetic, $f(100)=2^{100}-2 \bmod{100}$. $2^{10}=1024$, so $2^{100}-2\equiv 24^{10}-2\equiv (2^3 \cdot 3)^{10} - 2$ $\equiv 1024^3 \cdot 81 \cdot 81 \cdot 9 - 2 \equiv 24^3 \cdot 19^2 \cdot 9 - 2$ $\equiv 74\bmod{100} \Rightarrow \mathrm{(E)}$.
## Solution 3 (plain recurrence solving)
We derive the recurrence $S_{n+1}=2S_n + 2$ as above. Without guessing the form of the solution at this point we can easily solve this recurrence. Note that one can easily get rid of the "$+2$" as follows: Let $S_n=T_n-2$. Then $S_{n+1}=T_{n+1}-2$ and $2S_n+2 = 2(T_n-2)+2 = 2T_n-2$. Therefore $T_{n+1}=2T_n$. This obviously solves to $T_n=2^{n-1} T_1$. As $S_1=0$, we have $T_1=2$. Therefore $T_n=2^n$ and consecutively $S_n=2^n-2$. | {
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# Show $\int_0^{2\pi}\cos^2t\,dt=\pi$ using Pythagorean Theorem
The 'traditional' way I always see this integral calculated is with the identity
$$\cos^2t=\frac{1+\cos(2t)}{2}$$
My alternative method uses $$\cos^2t+\sin^2t=1$$. It's obvious that
$$\int_0^{2\pi}\cos^2t+\sin^2t\,dt=\int_0^{2\pi}1\,dt=2\pi$$
The interval of integration is an integer multiple of the periods of each function ($$\cos^2t$$ has a period of $$\pi$$), and so it seems reasonable to me that given the Pythagorean identity used above, $$\cos^2t$$ and $$\sin^2t$$ contribute, for lack of a better word, equally to this final answer of $$2\pi$$ above, and so the integral in the title should be half of $$2\pi$$, or just $$\pi$$.
Is this method valid? What additional statements, if any, are necessary to make it rigorous enough to be valid?
• One way to make it more rigorous is just to shift the integral for $\sin^2$ by $\pi/2$ so that you just get the integral for $\cos^2$ twice. – vrugtehagel May 8 at 16:58
• One of my high-school teachers showed me this trick for calculating the average of $\cos^2 x$, and it's stuck with me ever since. – Michael Seifert May 8 at 17:04
## 2 Answers
Your argument is definitely valid. To add more explanation, we can say that $$\int_{a}^{a+T} f(x)dx = \int_{0}^{T} f(x) dx$$ for any $$T$$-periodic function $$f(x)$$ (try to prove this rigorously), and then $$\int_{0}^{2\pi} \sin^{2} t dt = \int_{0}^{2\pi} \cos^{2}\left(t-\frac{\pi}{2}\right) \,dt = \int_{-\frac{\pi}{2}}^{2\pi - \frac{\pi}{2}} \cos^{2}t\,dt = \int_{0}^{2\pi} \cos^{2}t\,dt$$
Use $$\displaystyle\int_0^{2a}f(x)\ dx=2\int_0^af(x) \ dx$$ for $$f(2a-x)=f(x)$$
twice to find $$I=\int_0^{2\pi}\cos^2t\ dt=4\int_0^{\pi/2}\cos^2t\ dt$$
Now $$\displaystyle\int_a^bf(x) \ dx=\int_a^bf(a+b-x) \ dx$$
to find $$2\cdot\dfrac I4=\int_0^{\pi/2}(\cos^2t+\sin^2t)dt$$ | {
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# Teacher claims this proof for $\frac{\csc\theta-1}{\cot\theta}=\frac{\cot\theta}{\csc\theta+1}$ is wrong. Why?
My son's high school teacher says his solution to this proof is wrong because it is not "the right way" and that you have to "start with one side of the equation and prove it is equal to the other". After reviewing it, I disagree. I believe his solution is correct, even if not "the right way", whatever that means. I asked my son how he did it: he cross-multiplied the given identity, simplified it to a known/obvious equality, and then reversed the steps for the proof. This was a graded exam, and the teacher gave him a zero for this problem.
What do you think about my son's solution? Thanks!
Problem: prove the following trigonometric identity \begin{align*} \frac{\csc(\theta)-1}{\cot(\theta)}&=\frac{\cot(\theta)}{\csc(\theta)+1}\ .\\ \end{align*} Solution: for all real $$\theta$$ not equal to an integer multiple of $$\pi/2$$, we have \begin{align*} \cot^2(\theta)&=\cot^2(\theta)\\[8pt] \frac{\cos^2(\theta)}{\sin^2(\theta)}&=\cot^2(\theta)\\[8pt] \frac{1-\sin^2(\theta)}{\sin^2(\theta)}&=\cot^2(\theta)\\[8pt] \csc^2(\theta)-1&=\cot^2(\theta)\\[8pt] \frac{\csc^2(\theta)-1}{\cot(\theta)}&=\cot(\theta)\\[8pt] \frac{\csc(\theta)-1}{\cot(\theta)}&=\frac{\cot(\theta)}{\csc(\theta)+1} \end{align*} | {
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• Ask the professor? They may have a specific way they ask you to prove them. I have my students prove them a specific way. These can be googled or whatever. Feb 28, 2021 at 17:52
• Your son's proof is correct. He proved that the required identity is equivalent to $\cot^2 \theta = \cot^2 \theta$ which is true. So the identity is true. Feb 28, 2021 at 17:52
• You would be amazed how easy it is to get an b.s. in education in the US. Feb 28, 2021 at 17:53
• The proof looks good to me (and actually this is exactly the order I would have written it in). Sorry your son has a teacher who thinks math is just following a given series of steps and that there is one right way to do every problem. Feb 28, 2021 at 17:53
• The proof is correct and furthermore easy to follow. If the teacher believes it to be wrong, then they should be able to specifically name the error ("Line $n$ is wrong because..."). Saying that a method is "not the right way" is questionable pedagogy at best, and at worst, suggests that the teacher doesn't fully understand the material. But perhaps there is missing information, e.g., the full question might be, "Prove the statement using the Right Way method". Feb 28, 2021 at 18:04
I would at least have skipped the first line and started with $$\dfrac{\cos^2\theta}{\sin^2\theta} = \cot^2\theta.$$ | {
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One reason why an instructor might have qualms about this argument is the frequency with which students do this in the opposite direction, i.e. they write \require{cancel} \xcancel{ \begin{align} \frac{\csc(\theta)-1}{\cot(\theta)}&=\frac{\cot(\theta)}{\csc(\theta)+1} \\[8pt] \frac{\csc^2(\theta)-1}{\cot(\theta)}&=\cot(\theta)\\[8pt] \csc^2(\theta)-1&=\cot^2(\theta)\\[8pt] \frac{1-\sin^2(\theta)}{\sin^2(\theta)}&=\cot^2(\theta)\\[8pt] \frac{\cos^2(\theta)}{\sin^2(\theta)}&=\cot^2(\theta)\\[8pt] \cot^2(\theta)&=\cot^2(\theta) \end{align} } This would be logically correct if one wrote "This equality is true if the one below it is true." on each line. But one must be clear about the correct direction of "If ... then ...". Without those explicit words, the sequence of equalities can be taken to mean "If ... then ..." in just the opposite order.
Often what instructors want is something like this: $$\frac{\csc\theta-1}{\cot\theta} = \cdots\cdots\cdots\cdots = \frac{\cot\theta}{\csc\theta+1}.$$ in which each "equals" sign asserts the equality of things already known to be equal.
I would give full credit to the answer that your son wrote if each line had some brief explanation of how it is deduced from the line before it. For example, he could write this:
$$\cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta} = \frac{1-\sin^2\theta}{\sin^2\theta} = \csc^2\theta-1.$$ Then, dividing both sides of the equality $$\cot^2\theta = \csc^2\theta-1$$ by $$\cot\theta,$$ we get $$\cot\theta = \frac{\csc^2\theta-1}{\cot\theta}.$$ Finally, dividing both sides of that by $$\csc\theta+1$$ we get $$\frac{\csc\theta-1}{\cot\theta} = \frac{\cot\theta}{\csc\theta+1}.$$
Note that I wrote words above, not just mathematical notation. Well written answers do that, except perhaps in fairly simple cases.
• Very nice all the answer and the question +1 for all the users. Very nice the required cancel :-) for MathJAX. Feb 28, 2021 at 22:48
• @Sebastiano : Thank you. Mar 1, 2021 at 6:16 | {
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There are multiple motivations for writing proofs: perhaps to pass an exam (from a regular student's viewpoint), or perhaps to test understanding (from a teacher's perspective). Here, I'd like to highlight another important motivation: to enjoy the process of discovery.
Here's a geometric proof of the trigonometric identity. It is not practical to use in an exam, but it makes the abstract symbols more concrete. Coming up with multiple proofs of the same result can help one develop intuition and allow one to appreciate the connections between different areas of mathematics.
Problem statement: $$\frac{\csc\theta-1}{\cot\theta}=\frac{\cot\theta}{\csc\theta+1}$$
In this picture involving the unit circle, we construct some lengths which correspond to expressions in the problem statement. We start with $$\theta$$ and $$\phi = \frac{\pi}{2} - \theta$$.
Basic trigonometric definitions lead to the blue and orange lengths. By some angle-chasing (or a circle theorem), we have the $$\color{red}{\text{red}}$$ angle. Then, more angle-chasing gives:
The $$\color{red}{\text{red}}$$ angles are the same $$\implies\triangle ADB\sim \triangle BDE \implies \frac{AD}{DB}=\frac{DB}{ED}\iff$$ the problem statement.
• Love this geometric approach! If you don't mind, what do you use to produce such a nice and clear diagram of the geometry/trigonometry? Thanks! Feb 28, 2021 at 21:20
• Thank you. I'm glad you liked it. I used GeoGebra: geogebra.org/m/yngwjpcg Feb 28, 2021 at 21:34 | {
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I would assume that the teacher considered it an invalid answer, because your son assumed that the equality is true before showing that it is, if that even makes any sense. This is obviously a guess, but in the mind of the teacher, what he did was not "showing" that the equality is true, but merely "verifying" that it is. In other words, what the teacher likely wanted is for the students to get the answer on the right-hand side by exclusively using the left-hand side. More explicitly, even though the students knew what the end result should look like, they should have pretended they don't, manipulated the left-hand side and exclaimed "Eureka! It's true!" when they got the RHS.
With that logic, all he needed to do was to multiply numerator and denominator by $$\csc(\theta)+1$$, which immediately yields the desired result after simplifications. | {
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• Indeed, and a far more immediate solution! Feb 28, 2021 at 18:14
• Your answer reminds me of a high school maths teacher who said that I "thought too much like an engineer" (in the sense that I would often manipulate both sides into something that's true, in the style of the OP's son's proof). However, all of us think like this to a certain extent: some people are fast, and can immediately rewrite the proof (in their head) as "LHS = ... = ... = RHS"; some might hide that process on scratch paper. I don't really understand how one of these proofs can be better than the other, assuming they're correctly written. They all demonstrate the same math ability. Feb 28, 2021 at 18:25
• That is possible that is how the teacher thinks. Thanks for the reply! Feb 28, 2021 at 18:30
• Former high school math teacher here who taught students trig identity proofs, and this is exactly right. Moreover, it helps to keep them from making mistakes and incorrectly "proving" the statement after making a mistake on one side, then forcing another mistake on the other side to get to equality instead of searching for the problem in the work they've already done. It also helps them learn more about the relationships between different trig functions if they can only manipulate one side. Feb 28, 2021 at 18:31
• Further, the point of a trig identity problem isn't always just to demonstrate equality. There are typically multiple standards built into a single question. In this case, I'm guessing the teacher wanted something involving manipulating fractions. Feb 28, 2021 at 18:31
All of these methods of proof are valid, and should be graded as such (unless the question specifically said to use a certain method).
Method $$1$$ - Start with one side, simplify to the other. | {
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Method $$1$$ - Start with one side, simplify to the other.
$$\frac{\csc(\theta)-1}{\cot(\theta)}=\frac{(\csc(\theta)-1)(\csc(\theta)+1)}{\cot(\theta)(\csc(\theta)+1)}=\frac{\csc^{2}(\theta)-1}{\cot(\theta)( \csc(\theta)+1)}=\frac{\cot^{2}(\theta)}{\cot(\theta)(\csc(\theta)+1)}$$$$=\boxed{\frac{\cot(\theta)}{\csc(\theta)+1}}$$
Method $$2$$ - Start with the asserted identity, simplify to a known identity.
$$\frac{\csc(\theta)-1}{\cot(\theta)}=\frac{\cot(\theta)}{\csc(\theta)+1}\Longleftrightarrow(\csc(\theta)-1)(\csc(\theta)+1)=\cot^{2}(\theta)\Longleftrightarrow\csc^{2}(\theta)-1$$$$=\cot^{2}(\theta)\ \blacksquare$$
Method $$3$$ - Start with a known identity, turn it into the asserted identity (which is what your son did):
$$\csc^{2}(\theta)-1=(\csc(\theta)+1)(\csc(\theta)-1)=\cot^{2}(\theta)\Longrightarrow\boxed{\frac{\csc(\theta)-1}{\cot(\theta)}=\frac{\cot(\theta)}{\csc(\theta)+1}}$$
That said, however, perhaps the teacher took points off due to the length of your son's proof, which was more roundabout than it could have been. For instance, he could have started with a Pythagorean identity instead of starting with a '$$x=x$$' identity and using a Pythagorean identity, and took a few less steps in simplifying to the desired identity.
• +1 nice answer! Feb 28, 2021 at 18:29
It's disappointing to see that the teacher believes there is only one way to prove trigonometric identities. Actually, there are a number of approaches, all equally valid and useful in different circumstances. Sometimes it is more elegant to 'start with one side and prove that it is equal to the other', but a proof is a proof. | {
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Your son is actually using quite an advanced technique, where he shows that a statement $$P$$ implies another statement $$Q$$, and then shows that all of the steps are invertible, meaning that $$P \iff Q$$. As long as you are able to show that all of the steps are indeed invertible, this is a very useful method. Let's look at his method in more detail. He started by assuming that $$\frac{\csc \theta - 1}{\cot\theta} = \frac{\cot\theta}{\csc\theta+1}$$ and working from there. If we cross mutliply both sides, look at what happens: \begin{align} &\frac{\csc \theta - 1}{\cot\theta} = \frac{\cot\theta}{\csc\theta+1} \\[6pt] \implies & (\csc\theta-1)(\csc\theta+1) = \cot^2\theta \\[6pt] \implies & \csc^2\theta-1=\cot^2\theta \\[6pt] \implies & \frac{1-\sin^2\theta}{\sin^2\theta} = \frac{\cos^2\theta}{\sin^2\theta} \\[6pt] \implies & \frac{\cos^2\theta}{\sin^2\theta}=\frac{\cos^2\theta}{\sin^2\theta} \\[6pt] \implies & \cot^2\theta = \cot^2\theta \, . \end{align} Now, try reading the proof 'backwards'. Does $$\cot^2\theta = \cot^2\theta$$ imply $$\frac{\cos^2\theta}{\sin^2\theta} = \frac{\cos^2\theta}{\sin^2\theta} \, ?$$ Does $$\frac{\cos^2\theta}{\sin^2\theta} = \frac{\cos^2\theta}{\sin^2\theta}$$ imply $$\frac{1-\sin^2\theta}{\sin^2\theta} = \frac{\cos^2\theta}{\sin^2\theta} \, ?$$ Keep going. If the answer to all of these question is 'yes', then we have successfully proven that $$\frac{\csc \theta - 1}{\cot\theta} = \frac{\cot\theta}{\csc\theta+1} \iff \cot^2\theta = \cot^2\theta \, .$$ And since the RHS is an identity, the LHS must also be an identity. There's just one tiny snag to this proof. When we are going in a backwards direction, and conclude that $$(\csc\theta-1)(\csc\theta+1) = \cot^2\theta$$ implies $$\frac{\csc \theta - 1}{\cot\theta} = \frac{\cot\theta}{\csc\theta+1} \, ,$$ we are assuming that both $$\cot\theta$$ and $$\csc\theta + 1$$ are non-zero. But, your son directly addresses this by stating at the start of the proof, 'for all real $$\theta$$ not | {
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your son directly addresses this by stating at the start of the proof, 'for all real $$\theta$$ not equal to an integer multiple of $$\pi/2$$...', meaning that this is never a problem. This kind of attention to detail is very impressive for a high-school student. | {
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# [SOLVED]Sum with Binomial Coefficients
#### VincentP
##### New member
I'm having trouble proving the following identity (I don't even know if it's true):
$$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\binom{n-1}{k-1}$$ $$\forall n,k \in \mathbb{N} : n>k$$
Thank you in advance for any help!
Vincent
#### Prove It
##### Well-known member
MHB Math Helper
I'm having trouble proving the following identity (I don't even know if it's true):
$$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\binom{n-1}{k-1}$$ $$\forall n,k \in \mathbb{N} : n>k$$
Thank you in advance for any help!
Vincent
Maybe expanding the binomial coefficients would help. Recall that $\displaystyle {n\choose{m}} = \frac{n!}{m!(n-m)!}$
#### VincentP
##### New member
Maybe expanding the binomial coefficients would help. Recall that $\displaystyle {n\choose{m}} = \frac{n!}{m!(n-m)!}$
Well I have tried that of course:
$$\sum_{r=1}^k \frac{k!}{r!(k-r)!} \frac{(n-k-1)!}{(r-1)!(n-k-r)!} \stackrel{?}{=} \frac{(n-1)!}{(k-1)!(n-k)!}$$
But I don't know where to go from here since I still can't sum the left hand side. I also tried to prove it by induction but I failed to prove the induction step.
#### Sudharaka
##### Well-known member
MHB Math Helper
I'm having trouble proving the following identity (I don't even know if it's true):
$$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\binom{n-1}{k-1}$$ $$\forall n,k \in \mathbb{N} : n>k$$
Thank you in advance for any help!
Vincent
Hi Vincent,
Thank you for submitting this problem, I enjoyed solving it.
Since, $$\binom{k}{r}=\binom{k}{k-r}$$ we have,
$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\sum_{r=1}^{k}\binom{k}{k-r} \binom{n-k-1}{r-1}$
Using the Pascal's rule we get,
\begin{eqnarray}
\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}&=&\sum_{r=1}^{k}\left[{k+1 \choose k-r+1}-{k \choose k-r+1}\right]\binom{n-k-1}{r-1}\\
&=&\sum_{r=1}^{k}{k+1 \choose k-r+1}\binom{n-k-1}{r-1}-\sum_{r=1}^{k}{k \choose k-r+1}\binom{n-k-1}{r-1}\\
\end{eqnarray} | {
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\end{eqnarray}
Now use the Vandermonde's identity to get,
$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}={n \choose k}-\binom{n-1}{k}$
Using the Pascal's rule again we get,
$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}={n-1 \choose k-1}$
Kind Regards,
Sudharaka.
#### Opalg
##### MHB Oldtimer
Staff member
For a completely different, combinatorial, way to look at this problem, suppose that you have $n-1$ objects, and you want to select $k-1$ of them. There are $n-1\choose k-1$ ways of making the selection.
Now suppose that $k$ of the objects are white, and the remaining $(n-1)-k$ are black. Then another way to select $k-1$ objects is as follows. First, choose a number $r$ between 1 and $k$ (inclusive). Then select $r-1$ black balls and $(k-1)-(r-1) = k-r$ white balls. The number of ways to do that is ${k\choose k-r}{(n-1)-k\choose r-1} = {k\choose r}{n-k-1\choose r-1}.$ Sum that from $r=1$ to $k$ to get the total number of ways to select $k-1$ objects.
#### VincentP
##### New member
@ Sudharaka
Thank you very much for your explanation!
I have one question though: Doesn't Vandermonde's identity require the sum to start at r=0?
@Opalg
Thank you for your reply, that's a very interesting approach to the problem!
#### Sudharaka
##### Well-known member
MHB Math Helper
@ Sudharaka
Thank you very much for your explanation!
I have one question though: Doesn't Vandermonde's identity require the sum to start at r=0?
@Opalg
Thank you for your reply, that's a very interesting approach to the problem!
You are welcome. I have neglected an in between step that may have aroused the confusion.
$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\sum_{r=1}^{k}{k+1 \choose k-r+1}\binom{n-k-1}{r-1}-\sum_{r=1}^{k}{k \choose k-r+1}\binom{n-k-1}{r-1}$
Substitute $$u=r-1$$. Then the right hand side becomes,
$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\sum_{u=0}^{k}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k}{k \choose k-u}\binom{n-k-1}{u}$
I hope this clarifies your doubts. | {
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I hope this clarifies your doubts.
Kind Regards,
Sudharaka.
#### VincentP
##### New member
You are welcome. I have neglected an in between step that may have aroused the confusion.
$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\sum_{r=1}^{k}{k+1 \choose k-r+1}\binom{n-k-1}{r-1}-\sum_{r=1}^{k}{k \choose k-r+1}\binom{n-k-1}{r-1}$
Substitute $$u=r-1$$. Then the right hand side becomes,
$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\sum_{u=0}^{k}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k}{k \choose k-u}\binom{n-k-1}{u}$
I hope this clarifies your doubts.
Kind Regards,
Sudharaka.
Well not quite.
If you substitute the index of summation $u=r-1$ you have to change the lower as well as the upper bound of summation, because otherwise you change the number of summands. Therefore if you substitute $u=r-1$ we get:
$$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\sum_{u=0}^{k-1}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k-1}{k \choose k-u}\binom{n-k-1}{u}$$ Which doesn't match Vandermonde's Identity anymore, because the upper bound of summation doesn't appear in the lower index of the binomial coefficant.
Kind Regards,
Vincent
#### Sudharaka
##### Well-known member
MHB Math Helper
Well not quite.
If you substitute the index of summation $u=r-1$ you have to change the lower as well as the upper bound of summation, because otherwise you change the number of summands. Therefore if you substitute $u=r-1$ we get:
$$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\sum_{u=0}^{k-1}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k-1}{k \choose k-u}\binom{n-k-1}{u}$$ Which doesn't match Vandermonde's Identity anymore, because the upper bound of summation doesn't appear in the lower index of the binomial coefficant.
Kind Regards,
Vincent
Note that,
$\sum_{u=0}^{k-1}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k-1}{k \choose k-u}\binom{n-k-1}{u}=\sum_{u=0}^{k}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k}{k \choose k-u}\binom{n-k-1}{u}$ | {
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since when $$u=k$$ the right hand side is equal to zero. If you have any more questions about this please don't hesitate to ask.
Kind Regards,
Sudharaka.
#### VincentP
##### New member
Note that,
$\sum_{u=0}^{k-1}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k-1}{k \choose k-u}\binom{n-k-1}{u}=\sum_{u=0}^{k}{k+1 \choose k-u}\binom{n-k-1}{u}-\sum_{u=0}^{k}{k \choose k-u}\binom{n-k-1}{u}$
since when $$u=k$$ the right hand side is equal to zero. If you have any more questions about this please don't hesitate to ask.
Kind Regards,
Sudharaka.
I think that clarifies everything, thanks so much.
Vincent
#### Sudharaka
##### Well-known member
MHB Math Helper
I think that clarifies everything, thanks so much.
Vincent
I am glad to be of help.
Kind Regards,
Sudharaka. | {
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# Directional Derivative help, solving for derivative = 0 when given constants
A function that is useful in studying the air flow over mountains is
$$h(x,y) = \frac{h_0}{[(\frac{x}{a})^2+(\frac{y}{b})^2+1]^\frac{3}{2}}$$
where $h_0$, a, and b are all positive constants.
(a) Find $\nabla h$.
(b) Find the directional derivative of the point (x,y) in the direction of the vector $v=(v_1,v_2)$
(c) At what point(s) is the directional derivative equal to zero?
For part a), I found that $\nabla h$ is equal to $$\frac{-3h_0}{[(\frac{x}{a})^2+(\frac{y}{b})^2+1]^\frac{5}{2}}\langle\frac{x}{a^2},\frac{y}{b^2}\rangle$$
Using this, I found that for part b), I got that $D_uh(x,y) = \frac{-3h_0}{[(\frac{x}{a})^2+(\frac{y}{b})^2+1]^\frac{5}{2}}\langle\frac{x}{a^2},\frac{y}{b^2}\rangle \bullet\frac{1}{\sqrt{v_1^2+v_2^2}}\langle v_1, v_2 \rangle$
which simplifies to $D_uh(x,y) = \frac{-3h_0}{[(\frac{x}{a})^2+(\frac{y}{b})^2+1]^\frac{5}{2}{\sqrt{v_1^2+v_2^2}}}(\frac{xv_1}{a^2}+\frac{yv_2}{b^2})$
First of all, is this correct?
Secondly, given that I am correct up to this point, I am unsure on how to do part c). How can I solve for an x and y where $D_uh = 0$ when I do not have values for $v_1$ and $v_2$?
Any help here is appreciated. | {
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Any help here is appreciated.
• I suposse you have to solve the equation $\frac{xv_{1}}{a^{2}}+\frac{yv_{2}}{b^{2}}=0$ – julio godoy Jun 23 '16 at 6:36
• and the vector solution is $(-ya^{2},xb^{2})$, if the vector solution has unit lenght you have to normalize that vector – julio godoy Jun 23 '16 at 6:44
• @juliogodoy Wouldn't that solution require that $v_1 = -v_2$? if you plug in those x and y values in the equation, you get $-yv_1 + xv_2 = 0$. What am I missing here? – dibdub Jun 23 '16 at 6:46
• you can use also the fact that $u$ and $v$ are perpendicular then $u\cdot v = 0$ and if $u = (u_1,u_2)$ then $v=(-u_2,u_1)$ – Navaro Jun 23 '16 at 6:51
• In that case wouldnt the answer be $(\frac{-yv_2}{b^2},\frac{xv_1}{a^2})$? *edit: Actually it would be $(\frac{-v_2}{b^2},\frac{v_1}{a^2})$. Is that correct? – dibdub Jun 23 '16 at 6:54
The first two answers are correct.
For the third question: the directional derivative is equal to $0$ if $v =(v_1,v_2)$ is perpendicular to $\nabla h$ , so: $$v\cdot \nabla h = 0 \Rightarrow \dfrac{v_1}{a^2}x+\dfrac{v_2}{b^2}y = 0 \Rightarrow y = -\left( \dfrac{b}{a} \right)^2\left(\dfrac{v_1}{v_2}\right)x\Rightarrow y=cx \quad\text{where } c=-\left( \dfrac{b}{a} \right)^2\left(\dfrac{v_1}{v_2}\right)$$ which is the equation of a straight line passing through the origin $O=(0,0)$
conclusion: every point $(x,y)$ satisfying the previous equation makes the directional derivative in the direction of $v=(v_1,v_2)$ equal to zero
• Perfect, that makes sense. Thanks for all your help! – dibdub Jun 23 '16 at 7:47
The condition $x\frac{v_1}{a^2}+y\frac{v_2}{b^2}=0$ could be written $(P-O)\cdot\mathbf{u}=0$, where $P=(x,y)$, $O=(0,0)$ and $\mathbf{u}=\langle\frac{v_1}{a^2},\frac{v_2}{b^2}\rangle$.
This is the equation of the straight line passing through $O$ and orthogonal to $\mathbf{u}$.
The directional derivative is $0$ on each point of this line. | {
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The directional derivative is $0$ on each point of this line.
• Could you elaborate on this? Why is it (P-O)? What does P represent? How would I present my answer for part c) then? – dibdub Jun 23 '16 at 6:45
• @dubbler26: edited. – enzotib Jun 23 '16 at 7:03 | {
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# Calculating expectation of a function of a continuous random variable?
Assume we have a continuous random variable $$X$$, with a probability density function
$$f(x) = 1 - \frac{x}{2},\; 0 < x < 2$$
Let's say I want to calculate $$E(2X)$$. I will use two different methods, and get different results, so I obviously understand something wrong.
METHOD 1:
Using the formula for the expectation of a function of a random variable, I get:
$$E(2X) = \int_{-\infty}^{+\infty} 2xf(x)dx = \int_0^2 (2x-2x^2)dx = \frac{4}{3}$$
I understand this formula intuitively, as it's pretty much exactly the same as in the discrete case, if we replace the integral with a finite sum and the probability density function with the probability mass function. Only the values change, and not the distribution, so we only apply the function 2x to the values, and not the distribution $$f(x)$$.
METHOD 2:
Let's say I declare a new continuous random variable, $$Y = 2X$$. I can calculate the probability density function of $$Y$$:
$$g(y) = |\frac{dx}{dy}|f(x) = \frac{1}{2}f(\frac{y}{2}) = \frac{1}{2}(1-\frac{y}{4})$$
for $$0.
Then I can calculate:
$$E(2X) = E(Y) = \int_{-\infty}^{+\infty} y\;g(y)dx = \int_0^4 \frac{1}{2}(1-\frac{y}{4})dx = \frac{2}{3}$$
I don't understand what I did wrong here.
But I also don't understand the formula for $$g(y)$$ intuitively - shouldn't $$g(y)$$ and $$f(x)$$ be the same functions if they're injective, similar to the discrete case?
Anyway, if someone can point out what I did wrong, I would be very grateful. | {
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Anyway, if someone can point out what I did wrong, I would be very grateful.
You forgot to multiply by $$y$$ for the second computation.
$$g(y)$$ and $$f(x)$$ are different functions, but they are related by $$g(y)=\frac{1}{2}f\big(\frac{y}{2}\big)$$ and are both decreasing (also see here). For instance, $$h_{1}(x)=x$$ and $$h_{2}(x)=2x$$ are injective, but $$h_{1}(1)=1$$ and $$h_{2}(1)=2$$.
It is a good idea to check if $$g(y)=\frac{1}{2}(1-\frac{y}{4})$$ for $$0 is a valid pdf. Indeed we have $$g(y)\geq 0$$ for $$0 and $$\int_{-\infty}^{\infty}g(y)dy=\frac{1}{2}\int_{0}^{4}(1-\frac{y}{4})dy=1,$$ so $$g(y)$$ is a valid pdf.
Next we have $$E(2X) = E(Y) = \int_{-\infty}^{+\infty} y\;g(y)dx = \int_0^4 \frac{\color{red}{y}}{2}(1-\frac{y}{4})dx = \frac{4}{3}$$ | {
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# Is there a bijective map from $(0,1)$ to $\mathbb{R}$?
I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
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Yes, and there's even one from (0,1) to R²... – Axel Sep 21 '12 at 12:56
Here is a nice one ${}{}{}{}{}{}{}{}{}$, can you find the equation?
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That's really nice! – Clive Newstead Sep 21 '12 at 12:26
Note: I would be interested in an explicit formula. – The Chaz 2.0 Oct 21 '13 at 19:59
@The Chaz 2.0 If the diameter of the circle is 1 and the interval is $(0, 1)$ and 1/2 mapped to 0 and the distance between the center of the circle and the 2nd numberline is 1/2 then the formula is $f(x)=\frac{x-1/2}{\sqrt{x-x^2}}$. Becouse scaling does not matter you can multiply by 2 and get the formula $f(x)=\frac{2x-1}{\sqrt{x-x^2}}$, but i don't know how you can get Cameron Buie's formula, out of this. – 05storm26 Feb 2 '14 at 8:31
I am thinking about: $f:(0,1)\to\mathbb{R}$, $f(x)=-\cot\left(\pi x\right)$ – Darius Apr 22 '14 at 19:58
As far as I can tell, the connection between our answers' formulae is coincidental (aside from the fact that if $g$ is a positive function on some subset of $\Bbb R,$ then so is $\sqrt{g},$ so our functions are related, but not in an obvious heuristic way. Please let me know if you can discern a connection between our answers (aside from the obvious algebraic one). – Cameron Buie Oct 31 '14 at 3:07
Here is a bijection from $(-\pi/2,\pi/2)$ to $\mathbb{R}$: $$f(x)=\tan x.$$ You can play with this function and solve your problem.
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$g(x)=\frac 1{1+e^x}$ gives a bijection from $\Bbb R$ to $(0,1)$, so take the inverse of this map.
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And the inverse is $f(x)=\ln\left(\frac{1}{x}-1\right)$ – celtschk Sep 21 '12 at 12:19
A homeomorphism (continuous bijection with a continuous inverse) would be $f:(0,1)\to\Bbb R$ given by $$f(x)=\frac{2x-1}{x-x^2}.$$ | {
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Added: Let me provide some explanation of how I came by this answer, rather than simply leave it as an unmotivated (though effective) formula and claim in perpetuity.
Many moons ago, I was assigned the task of demonstrating that the real interval $(-1,1)$ was in bijection with $\Bbb R.$ Prior experience with rational functions had shown me graphs like this:
The above is a graph of a continuous function from most of $\Bbb R$ onto $\Bbb R.$ This doesn't do the trick on its own, since it certainly isn't injective. However, it occurred to me that if we restrict the function to the open interval between the two vertical asymptotes, we get this graph, instead:
This graph is of a continuous, injective (more precisely, increasing) function from a bounded open interval of $\Bbb R$ onto $\Bbb R.$ This showed that rational functions could do the job. Other options occurred to me, certainly (such as trigonometric functions), but of the ideas I had (and given the results I was allowed to use) at the time, the most straightforward approach turned out to be using rational functions.
Now, given the symmetry of the interval $(-1,1)$ (and, arguably, of $\Bbb R$) about $0,$ the natural choice of the unique zero of the desired function was $x=0.$ In other words, I wanted $x$ to be the unique factor of the desired rational function's numerator that could be made equal to $0$ in the interval $(-1,1)$--meaning that for $\beta\in(-1,1)$ with $\beta\ne0,$ I needed to make sure that $x-\beta$ was not a factor of the numerator. For simplicity, I hoped that I could make $x$ the only factor of the numerator that could be made equal to $0$ at all--that is, I hoped that I could have $\alpha x$ as the numerator of my function for some nonzero real $\alpha.$ | {
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In order to get the vertical asymptotic behavior I wanted on the given interval--that is, only at the interval's endpoints--I needed to make sure that $x=\pm1$ gave a denominator of $0$--that is, that $x\mp1$ were factors of the denominator--and that for $-1<\beta<1,$ $\beta$ was not a zero of the denominator--that is, that $x-\beta$ wasn't a factor of the denominator. For simplicity, I hoped that I could make $x\mp1$ the only factors of the denominator.
Playing to my hopes, I assumed $\alpha$ to be some arbitrary nonzero real, and considered the family of functions $$g_\alpha(x)=\frac{\alpha x}{(x+1)(x-1)}=\frac{\alpha x}{x^2-1},$$ with domain $(-1,1).$ It was readily seen that all such functions are real-valued and onto $\Bbb R.$
I wanted more, though! (I'm demanding of my functions when I can be. What can I say?) I wanted an increasing function. I determined (through experimentation which suggested proof) that $h_\alpha$ would be increasing if and only if $\alpha<0.$ Again, for convenience, I chose $\alpha=-1,$ which gave me the function that satisfied the desired (and required) properties: $g:(-1,1)\to\Bbb R$ given by $$g(x)=\frac{-x}{x^2-1}=\frac{x}{1-x^2}.$$
Much later, you posted your question, and I realized (again, based on experience) that my earlier result could be adapted to the one you wanted. Playing around a bit with linear interpolation showed that the function $h:(0,1)\to(-1,1)$ given by $h(x)=2x-1$ was a bijection--in fact, an increasing bijection. | {
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It is readily shown (and I had previously seen) that if $X,Y,$ and $Z$ are ordered sets and if we are given increasing maps $X\to Y$ and $Y\to Z,$ then the composition of those maps is an increasing map $X\to Z.$ Also, it is readily shown (and I had seen previously) that if both such maps are continuous and surjective, then so is their composition. Just from these results, my originally posted map was obtained (though named differently): \begin{align}(g\circ h)(x) &= g\bigl(h(x)\bigr)\\ &= \frac{h(x)}{1-\left(h(x)\right)^2}\\ &=\frac{2x-1}{1-(2x-1)^2}\\ &=\frac{2x-1}{1-\left(4x^2-4x+1\right)}\\ &= \frac{2x-1}{4x-4x^2}\\ &=\frac{2x-1}{4(x-x^2)}.\end{align}
As lhf astutely pointed out shortly thereafter (and as I should have seen immediately), the factor of $4$ in the denominator serves no particular purpose, hence its later removal to yield the function $f$ that I eventually posted.
The remaining claim that I made (that $f$ has a continuous inverse), I leave to you (the reader). If you're curious how I determined this, try to prove it on your own first. If you're stymied (or if you simply want to run your proof attempt by me), let me know. I will do what I can to get you "unstuck."
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This is the answer you want. A direct map that just stretches the unit interval onto the (-inf,inf) interval. This shows you don't need an exotic function like the bijection between [0,1] and [0,1]^2. – John Baber-Lucero Sep 21 '12 at 12:17
No need for that 4... – lhf Sep 21 '12 at 12:22
Fair point, @lhf. Not sure why I bothered with that.... – Cameron Buie Sep 21 '12 at 17:21
+1 for the motivation. – LePressentiment Jan 6 at 4:45
For a less differentiable example, consider the bijection in the following picture, | {
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For a less differentiable example, consider the bijection in the following picture,
In symbols, given $x \in (0,1)$ let $n$ be the largest natural number such that $1-\frac{1}{n}<x$, define $$y=\frac{x-n}{\frac{1}{n}-\frac{1}{n+1}}$$ to be the renormalized version of $x$ if the interval $(1-\frac{1}{n},1-\frac{1}{n+1}]$ is rescaled and shifted to map to $(0,1)$. Then we have the following bijection: $$f(x)=\begin{cases}\frac{n-1}{2}+y,& n \text{ odd} \\ -\frac{n-2}{2}-y,& n \text{ even}\end{cases}$$
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Not only less differentiable but also much less continuous. – Stefan Geschke Sep 21 '12 at 10:15
(: One might even consider the composition $f \circ g$ with a nowhere continuous bijection $g:(0,1)\rightarrow (0,1)$. – Nick Alger Sep 21 '12 at 10:27
Yes. let $f(x)=\tan((x-1/2)\pi)$. the domain is $(0,1)$ and range is $\mathbb{R}$
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Here is a more visual description of the same function (modulo a factor of $\pi$) Bend the line segment (0, 1) into a semicircle, with the open part facing upwards, and rest it on the real line (with 0.5 on the semicircle resting on 0 on the real line). To map a point on one line to a point on the other, draw a line through that point and the centre of the circle. The mapped point is where it intersects the other line. – Max Sep 21 '12 at 10:10
Of course, knowing the domain and range simply guarantees that $f$ is surjective, but this $f$ does turn out to be injective, too. – Cameron Buie Sep 21 '12 at 10:36 | {
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Yes, see above answers. There are even bijective maps between $(0,1)$ and $\mathbb{R}^n$. To see this, note that a bijection $\phi$ between $(0,1)$ and $(0,1)^2$ can be made in this way: Let $x= 0.b_1b_2\ldots$, with $b_j$ being the digits in a decimal expansion. Define $$\phi(x) = (0.b_1b_3b_5\ldots,0.b_2b_4b_6\ldots),$$ i.e., extract even and odd digits. For $\phi^{-1}(x_1,x_2)$, let $x_1 = 0.a_1a_2a_3\ldots$, and $x_2=b_1b_2b_3\ldots$. Then, $$\phi^{-1}(x,y) = 0.a_1b_1a_2b_2\cdots$$ Some care has to be taken with identification between digital expansions like $0.199999\cdots$ and $0.20000\cdots$, but that is an exercise.
Having the bijection between $(0,1)$ and $(0,1)^2$, we can apply one of the other answers to create a bijection with $\mathbb{R}^2$.
The argument easily generalizes to $\mathbb{R}^n$.
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Actually you don't even have to generalize the argument: If you have the bijection between $(0,1)$ and $(0,1)^2$, you get a bijection from $(0,1)$ to $(0,1)^3$ by just applying the same bijection to one of the two factors of $(0,1)^2$. Of course the same way you get to $(0,1)^n$. – celtschk Sep 21 '12 at 12:16
The trigonometric function $\tan x$ is an invertible function from $(-\pi/2,\pi/2)$ to $\mathbb{R}$. Also to find an invertible function from $(0,1)$ to $(-\pi/2,\pi/2)$ find the equation of the straight line joining the points $(0,-\pi/2)$ and $(1,\pi/2)$. Now compose the two functions together. You can likewise find bijections between any two open intervals and any open interval and $\mathbb{R}$.
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$x \mapsto \ln (- \ln x)$ with the inverse $y \mapsto e^{-e ^ {\ y}}$. It's also a $C ^ \infty$ diffeomorphism.
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By virtue of http://natureofmathematics.wordpress.com/lecture-notes/cantor/, here's another picture. The interval at the bottom is $\mathbb{R}$. | {
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It's not clear to me which point on the circle is missed from the map described by the blue arrows. It looks like it could be some point between the images of the point $0$ and the point $1$, but then the map described by the red arrows from the circle to the real line is not a bijective map because that missed point has no preimage and so its image in the real line after the red mapping has no preimage in the interval. This is a poor picture. – Dan Rust Feb 18 '14 at 12:45
the idea is good : draw a circle of radius $1$ and center $(0,1)$, draw the segment $(-1,1) , (1,1)$, and map the segment to $\mathbb{R}$ by taking from any point of the segment the nearest point on the half bottom circle, drawing the line passing through these 2 points, and considering its intersection with the horizontal line $y = 0$. finally, prove that every continuous bijection between $]-1;1[$ and $\mathbb{R}$ can be constructed this way (replacing the circle by other curves). and the reciprocal : characterize the curves from which we get such a bijection. – user1952009 Jan 5 at 22:54 | {
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# Determining compositions of trig functions by knowing Euler's identity etc
How does one determine: $$\cos^2(\arctan(x))?$$
I know what it is equal to, since its in the tables. But without working with many trigonometric identities, its not clear how to find such things.
How would you see this with the minimal number of trig identities?
$\cos^2(\arctan(x))=\cos(\arctan(x))\cos(\arctan(x))=\frac{1}{\sqrt{1+x^2}}\frac{1}{\sqrt{1+x^2}}=\frac1{1+x^2}.$
The one identity I used here, I didn't know. It seems in similar situations, on an exam, I would have massive trouble without these identities. Can all of these sorts of things be solved by knowing something about Eulers identity and such?
Define the point $P(\theta)=(1,x)$ so that $\tan(\theta) = x$. Note that we need to have $\theta$ in the first or fourth quadrant in order for $\theta = \arctan x$ to be true.
From the picture we see that $\cos^2(\arctan x) = \cos^2 \theta = \dfrac{1}{1+x^2}$.
I find that the easiest way to solve these is to let $\theta$ be $\arctan{x}$, so that $\tan{\theta}=x$. Then we have \begin{align*} \frac{\sin{\theta}}{\cos{\theta}}&=x\\ \cos{\theta}&=\frac{\sin{\theta}}{x}\\ \cos{\theta}&=\frac{\sqrt{1-\cos^2{\theta}}}{x}\\ x\cos{\theta}&=\sqrt{1-\cos^2{\theta}}\\ x^2\cos^2{\theta}&=1-\cos^2{\theta}\\ (x^2+1)\cos^2{\theta}&=1\\ \cos^2{\theta}&=\frac{1}{x^2+1}\\ \end{align*} This may look tougher, but after the third step, we only need to use algebraic manipulations, and don't need to worry about trigonometry anymore. The two nicest ways to make a trig problem easier to solve are getting rid of inverses like I have in this problem, and writing everything in terms of sine and cosine. | {
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• Is $\sin\theta$ always non negative like $$\sqrt{1-\cos^2\theta}$$ Nov 13 '17 at 4:08
• Best way to fix that problem is to already square after the second step. Nov 13 '17 at 6:55
• I think that the OP main concern is not to solve the problem posted. That is just an example. He seems to look for a general way of solving trig functions with some sort of simplifying tool like for example Euler’s identity. Nov 13 '17 at 7:35
From the well-known trig identity $$\sec^2y=1+\tan^2y$$ and $$\sec y=\frac{1}{\cos y}$$ one can easily find $$\cos^2y=\frac{1}{1+\tan^2y}$$ Plugging $y=\arctan x$ implies $$\cos^2(\arctan x)=\frac{1}{1+x^2}$$
Firstly, lets check the range of commonly defined $\arctan (x)$ because our result will depend on this. It is generally taken to be $(-\frac{\pi}{2}, \frac{\pi}{2})$. So $\cos$ will be positive over this domain.
Here this analysis is not of much use as we have squared the $\cos$ term, which will always be positive anyway.
Now to get the value of $\cos(\arctan(x))$ for acute angle $\arctan(x)$, let this angle be $\phi$.
\begin{align} \phi &= \arctan (x) \tag{1}\\ \tan(\phi) &= x\\ \frac{1}{1+\tan^2 \phi} &= \frac{1}{1+x^2}\\ \cos^2 \phi &= \frac{1}{1+x^2}\\ \implies \cos^2 (\arctan x) &=\frac{1}{1+x^2} \end{align}
There are (as is often the case with trigonometric identities) several ways to solve this, and the already-posted answers are just fine. Here's another, which I like because it uses a very general-purpose tool.
The tool in question: the famous half-angle formulae. These are extremely useful and reduce many trigonometric identities to routine algebraic manipulation. In case you don't already know them, here they are. Let $t=\tan\frac12\theta$; then $\cos\theta=\frac{1-t^2}{1+t^2}$, and $\sin\theta=\frac{2t}{1+t^2}$, and $\tan\theta=\frac{2t}{1-t^2}$. | {
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Why are these useful here? Because $\cos^2\phi$ is very closely related to $\cos2\phi$, which means we have cosine of twice an arctangent, which if you think about it for a moment you'll see is exactly what the half-angle formula for cos tells us how to handle.
So. We're taking $\tan^{-1}x$ and we want (in effect) the cosine of twice this, so write $x=\tan\frac12\theta$. The formula above says that $\cos\theta=\frac{1-x^2}{1+x^2}$; call this $c$. The thing we were actually asked for is $\cos^2\frac12\theta$ and we have $c=2\cos^2\frac12\theta-1$ so $\cos^2\frac12\theta=\frac{c+1}2=\frac1{1+t^2}$. | {
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# Find expression for sum of series
I want to find a formula for the sum of this series using its general term. How to do it?
Series $$S_n = \underbrace{1/3 + 2/21 + 3/91 + 4/273 + \cdots}_{n \text{ terms}}$$
General Term
$$S_n = \sum_{i=1}^{n} \frac{i}{i^4+i^2+1}$$
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I changed n\ \mbox{terms} to n\text{ terms}. The mismatch in sizes of fonts resulted from use of an incorrect method. – Michael Hardy Jul 12 '14 at 22:23
Using partial fractions, we get $\dfrac{i}{i^4+i^2+1} = \dfrac{\tfrac{1}{2}}{i^2-i+1} - \dfrac{\tfrac{1}{2}}{i^2+i+1}$.
Thus, $S_n = \displaystyle\sum_{i = 1}^{n}\dfrac{i}{i^4+i^2+1} = \dfrac{1}{2}\sum_{i = 1}^{n}\dfrac{1}{i^2-i+1} - \dfrac{1}{i^2+i+1}$.
Since $(i+1)^2-(i+1)+1 = i^2+i+1$, this sum telescopes to $S_n = \dfrac{1}{2}\left(1 - \dfrac{1}{n^2+n+1}\right)$.
Taking the limit as $n \to \infty$ gives $S = \dfrac{1}{2}$.
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thank you so much,but i have one doubt ,in which cases can i apply partial fraction to calculate sum? – user3481652 Jul 12 '14 at 20:47
Anytime you have a rational function (whose numerator has a smaller degree than its denominator), you can always try using partial fractions. Whether or not it will lead to a solution depends on the particular problem. – JimmyK4542 Jul 12 '14 at 20:49
okay thank you for your help :) – user3481652 Jul 12 '14 at 20:51
Note that the factorisation $i^4+i^2+1=(i^2+i+1)(i^2-i+1)$ comes easily if you notice that $(i^2-1)(i^4+i^2+1)=i^6-1=(i^3+1)(i^3-1)=(i+1)(i^2-i+1)(i-1)(i^2+i+1)$ – Mark Bennet Jul 12 '14 at 21:06
Use partial fractions,
$$\frac{i}{i^4+i^2+1} = \frac{1}{2(i^2-i+1)}-\frac{1}{2(i^2+i+1)}$$
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Write it out: | {
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$$\frac{i}{i^4+i^2+1} = \frac{1}{2(i^2-i+1)}-\frac{1}{2(i^2+i+1)}$$
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Write it out:
$$\begin{eqnarray} \sum_{i=1}^\infty \frac{i}{i^4 + i^2 + 1} &=& \sum_{i=1}^\infty \left( \frac{2}{4i^2 - 4i + 4} - \frac{2}{4i^2 + 4i + 4}\right)\\ &=& \sum_{i=1}^\infty \left( \frac{2}{\Big(2i-1\Big)^2 + 3} - \frac{2}{\Big( 2i + 1\Big)^2 + 3}\right)\\ &=& \frac{2}{\Big( 2 - 1\Big)^2 + 3} + \sum_{i=1}^\infty \left( \frac{2}{\Big(2i+1\Big)^2 + 3} - \frac{2}{\Big( 2i + 1\Big)^2 + 3}\right)\\ &=& \frac{1}{2}. \end{eqnarray}$$
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Mind to give reason for downvote? – johannesvalks Jul 12 '14 at 21:06
I didn't downvote you. But I think you posted the solution AFTER other users. I think they care about "speed" + "accuracy"....next time... – NotALoner Jul 12 '14 at 21:35
Well - that is the problem when you reply a post and in the mean time you watch the worldcup so you are late with the final result.... Thanks for the tip / advice!! – johannesvalks Jul 12 '14 at 21:39
@ user2584283 I think the OP asks for (a formula for) the sum of the series. – user84413 Jul 13 '14 at 1:01
If you look at the question's edit history, you will see that at one time, it asked for the sum of the infinite series, not just the first $n$ terms. – JimmyK4542 Jul 13 '14 at 1:41 | {
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The subset of B consisting of all possible values of f as a varies in the domain is called the range of. 1 T(~x + ~y) = T(~x) + T(~y)(preservation of addition) 2 T(a~x) = aT(~x)(preservation of scalar multiplication) Linear Transformations: Matrix of a Linear Transformation Linear Transformations Page 2/13. A basis for the kernel of L is {1} so the kernel has dimension 1. Matrix vector products as linear transformations. All Slader step-by-step solutions are FREE. Inversion: R(z) = 1 z. Let A = 2 4 0. Find the kernel of the linear transformation L: V→W. Proof: This theorem is a proved in a manner similar to how we solved the above example. range(T)={A in W | there exists B in V such that T(B)=A}. The $$\textit{nullity}$$ of a linear transformation is the dimension of the kernel, written $$nul L=\dim \ker L. Thus, we should be able to find the standard matrix for. It is essentially the same thing here that we are talking about. Let T: V !W be a linear transformation. It relates the dimension of the kernel and range of a linear map. In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (for example, two vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. Construct a linear transformation f and vector Y so that the system takes the form f(X)=Y. I tried to RREF it on my calculator but it says invalid dim, I am not sure of what to do and all the examples I have looked up are for square matrices, any help would be appreciated, thanks!. And we saw that earlier in the video. Suppose T : V !W is a linear transformation. In Section 4, we define the kernel whitening transformation and orthogonalize non-. Before we look at some examples of the null spaces of linear transformations, we will first establish that the null space can never be equal to the empty set, in. The confidence of the interval [107, 230] is less | {
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space can never be equal to the empty set, in. The confidence of the interval [107, 230] is less than 95%. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. The function of kernel is to take data as input and transform it into the required form. 0004 From the previous lesson, we left it off defining what the range of a linear map is. be a linear transformation. Specifically, if T: n m is a linear transformation, then there is a unique m n matrix, A, such that T x Ax for all x n. Griti is a learning community for students by students. Algebra Linear Algebra: A Modern Introduction 4th Edition In Exercises 5-8, find bases for the kernel and range of the linear transformations T in the indicated exercises. " • The fact that T is linear is essential to the kernel and range being subspaces. LTR-0060: Isomorphic Vector Spaces We define isomorphic vector spaces, discuss isomorphisms and their properties, and prove that any vector space of dimension is isomorphic to. (a) Find a basis of the range of P. the kernel of a a linear transformation is the set of vectors in the null space of the matrix for that linear transformation. Kernel, image, nullity, and rank Math 130 Linear Algebra D Joyce, Fall 2015 De nition 1. Linear Transformation. Since the nullity is the dimension of the null space, we see that the nullity of T is 0 since the dimension of the zero vector space is 0. The following examples illustrate the syntax. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. To prove the transformation is linear, the transformation must preserve scalar multiplication, addition, and the zero vector. We then consider invertible linear transformations, and then use the resulting ideas to prove the rather stunning result that (in a very precise sense). Vector Spaces and | {
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ideas to prove the rather stunning result that (in a very precise sense). Vector Spaces and Linear Transformations Beifang Chen Fall 2006 1 Vector spaces A vector space is a nonempty set V, whose objects are called vectors, equipped with two operations, called addition and scalar multiplication: For any two vectors u, v in V and a scalar c, there are unique vectors u+v and cu in V such that the following properties are satisfled. Kernel trick allows us to transform the data in high dimensional (potentially infinite) using inner products, without actually using the non linear feature mapping. SUBSCRIBE to the channel and. We discuss the kernel and range of linear transformations, and then prove that the range of a linear transformation is a subspace. What is the outcome of solving the problem?. The Gaussian is a self-similar function. Sums and scalar multiples of linear transformations. This MATLAB function returns predicted class labels for each observation in the predictor data X based on the binary Gaussian kernel classification model Mdl. Find the kernel of f. Then the Kernel of the linear transformation T is all elements of the vector space V that get mapped onto the zero element of the vector space W. In Section 3, we compute the whitening transformation. We show that for high-dimensional data, a particular framework for learning a linear transformation of the data based on the LogDet divergence can be efficiently kernelized to learn a metric (or equivalently, a kernel function) over an. T is a linear transformation. The problem comes when finding the Kernel basis. Define the linear transformation T(x) = A * x for A an m by n matrix. T(x 1,x 2,x 3,x 4)=(x 1−x 2+x 3+x 4,x 1+2x 3−x 4,x 1+x 2+3x 3. We denote the kernel of T by ker(T) or ker(A). Now, we connect together all the ideas we’ve talked. To find the null space we must first reduce the #3xx3# matrix found above to row echelon form. Linear transformations as matrix vector products. S: ℝ3 → ℝ3. Define | {
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above to row echelon form. Linear transformations as matrix vector products. S: ℝ3 → ℝ3. Define pre-image of U, denoted T -1. Griti is a learning community for students by students. One-to-One linear transformations: In college algebra, we could perform a horizontal line test to determine if a function was one-to-one, i. De nition 1. + for all vectors VI, for all scalars Cl, F(cv) for all scalars c, for all ve V, for all A function F: V —W is linear W be a subspace of Rk Let V be a subspace of Let it respects the linear operations,. linear transformation. Up Main page Definition. In particular, there exists a nonzero solution. Let V;W be vector spaces over a eld F. 2 The kernel and range of a linear transformation. Most off-the-shelf classifiers allow the user to specify one of three popular kernels: the polynomial, radial basis function, and sigmoid kernel. The same considerations apply to rows as well as columns. For example linear, nonlinear, polynomial, radial basis function. Recall that if a set of vectors v 1;v 2;:::;v n is linearly independent, that means that the linear combination c. The null space of T, denoted N(T), is de ned as N(T) := fv2V: T(v) = 0g: Remark 3. Let T: R n → R m be a linear transformation. Find the kernel of the linear transformation. If T : Rm → Rn is a linear transformation, then the set {x | T(x) = 0 } is called the kernelof T. To see why image relates to a linear transformation and a matrix, see the article on linear. Note that the squares of s add, not the s 's themselves. 2 Kernel and Range of a Linear Transformation Performance Criteria: 2. IV Image and Kernel of Linear Transformations Motivation: In the last class, we looked at the linearity of this function: F : R3 R2 F(x,y,z)=(x+y+z,2x-3y+4z) How does a 3 dimensional space get ‘mapped into’ a 2 dimensional space? At least one dimension ‘collapses’, or disappears. Let T: V !Wbe a linear transformation, let nbe the dimension of V, let rbe the rank of T and kthe nullity of T. | {
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a linear transformation, let nbe the dimension of V, let rbe the rank of T and kthe nullity of T. Because Tis one-to-one, the dimension of the image of Tmust be n. Gaussian Kernel always provides a value between 0 and 1. Linear Transformations Find the Kernel The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre- image of the transformation ). Corollary 2. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. For instance, for m = n = 2, let A = • 1 2 1 3 ‚; B = • 2 1 2 3 ‚; X = • x1 x2 x3 x4 ‚: Then F: M(2;2)! M(2;2) is given by F(X) = • 1 2 1 3 ‚• x1 x2 x3 x4 ‚• 2 1 2 3 ‚ = • 2x1 +2x2 +4x3 +4x4 x1 +3x2 +2x3 +6x4 2x1 +2x2 +6x3 +6x4 x1 +3x2 +3x3 +9x4 ‚: (b) The function D: P3! P2, deflned by D ¡ a0 +a1t+a2t 2 +a 3t 3 ¢ = a1 +2a2t+3a3t2; is a linear transformation. Notation: f: A 7!B If the value b 2 B is assigned to value a 2 A, then write f(a) = b, b is called the image of a under f. The dimension of the kernel of T is the same as the dimension of its null space and is called the nullity of the transformation. Support vector machine with a polynomial kernel can generate a non-linear decision boundary using those polynomial features. The offset c determines the x-coordinate of the point that all the lines in the posterior go though. Find the kernel of the linear transformation L: V→W. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The transformation defines a map from ℝ3 to ℝ3. (b) Find the matrix representation of L with respect to the standard basis 1;x;x2. Convolution with a Gaussian is a linear operation, so a convolution with a Gaussian kernel followed by a convolution with again a Gaussian kernel is equivalent to | {
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with a Gaussian kernel followed by a convolution with again a Gaussian kernel is equivalent to convolution with the broader kernel. First here is a definition of what is meant by the image and kernel of a linear transformation. Suppose L∶V → W is a linear isomorphism then it is a bijection. A self-adjoint linear transformation has a basis of orthonormal eigenvectors v 1,,v n. The kernel of 𝐴 is calculated by finding the reduced echelon form of this matrix using Gauss-Jordan elimination and then writing the solution in a particular way. RHS of equation is a 2 row by 1 column matrix. {\mathbb R}^n. Use the kernel and image to determine if a linear transformation is one to one or onto. (The dimension of the image space is sometimes called the rank of T, and the dimension of the kernel is sometimes called the nullity of T. Finding matrices such that M N = N M is an important problem in mathematics. Non Linear SVM using Kernel. Kernel algorithms using a linear kernel are often equivalent to their non-kernel counterparts, i. The kernel or null-space of a linear transformation is the set of all the vectors of the input space that are mapped under the linear transformation to the null vector of the output space. SVM algorithms use a set of mathematical functions that are defined as the kernel. T(v) = Av represents the linear transformation T. Let T: V !Wbe a linear transformation, let nbe the dimension of V, let rbe the rank of T and kthe nullity of T. Find a basis of the null space of the given m x n matrix A. De ne T : P 2!R2 by T(p) = p(0) p(0). 3, -3 , 1] Find the basis of the image of a linear transformation T defined by T(x)=Ax. the kernel of a a linear transformation is the set of vectors in the null space of the matrix for that linear transformation. Corollary 2. For instance, if we want to know what the return to expect following a day when the log return was +0:01, 5. A= 0 1 −1 0. Suppose T:R^3 \\to R^3,\\quad T(x,y,z) = (x + 2y, y + 2z, z + 2x) Part of Solution: | {
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5. A= 0 1 −1 0. Suppose T:R^3 \\to R^3,\\quad T(x,y,z) = (x + 2y, y + 2z, z + 2x) Part of Solution: The problem is solved like this: A =. ) T: R^2 rightarrow R^2, T(x, y) = (x + 2y, y - x) ker(T) = {: x, y R} T(v) = Av represents the linear transformation T. Because is a composition of linear transformations, itself is linear (Theorem th:complinear of LTR-0030). So,wehave w 1 = v1 kv1k = 1 √ 12 +12. The event times that satisfy include 107, 109, 110, 122, 129, 172, 192, 194, and 230. Find the matrix of the orthogonal projection onto W. Let T be a linear transformation from Rm to Rn with n × m matrix A. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It is essentially the same thing here that we are talking about. Sources of subspaces: kernels and ranges of linear transformations. be a linear transformation. Non Linear SVM using Kernel. TRUE Remember that Ax gives a linear combination of columns of A using x entries as weights. The null space of T, denoted N(T), is de ned as N(T) := fv2V: T(v) = 0g: Remark 3. Affine transformations", you can find examples of the use of linear transformations, which can be defined as a mapping between two vector spaces that preserves linearity. Find the kernel of the linear transformation. The following is a basic list of model types or relevant characteristics. SUBSCRIBE to the channel and. 2 (The Kernel and Range)/3. These are all vectors which are annihilated by the transformation. Then rangeT is a finite-dimensional subspace of W and dimV = dimnullT +dimrangeT. (f) The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation. Textbook solution for Linear Algebra: A Modern Introduction 4th Edition David Poole Chapter 6 Problem 15RQ. 2-T:R 3 →R 3,T(x,y,z)=(x,0,z). Introduction to Linear Algebra exam problems and solutions at the Ohio | {
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3 →R 3,T(x,y,z)=(x,0,z). Introduction to Linear Algebra exam problems and solutions at the Ohio State University. 4 LECTURE 7: LINEAR TRANSFORMATION We have L(v) = 0W = L(0V). 2 Kernel and Range of linear Transfor-mation We will essentially, skip this section. Find the matrix of the given linear transformation T with respect to the given basis. Here we consider the case where the linear map is not necessarily an isomorphism. Before we do that, let us give a few definitions. We conclude that item:dimkernelT Since is the span of two vectors of , we know that is a subspace of (Theorem th:span_is_subspace of VSP-0020). It doesn't hurt to have it, but it isn't necessary here (in finding the kernel). You can even pass in a custom kernel. Neal, WKU Theorem 2. For two linear transformations K and L taking Rn Rn , and v Rn , then in general K(L(v)) = L(K(v)). Section 2 describes the calculation of the canonical angles. In this section, you will learn most commonly used non-linear regression and how to transform them into linear regression. If there are page buffers, the total number of bytes in the page buffer area is 'data_len'. Find the matrix of the given linear transformation T with respect to the given basis. And we saw that earlier in the video. Find the kernel of the linear transformation. , the solutions of the equation A~x = ~ 0. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In the case where V is finite-dimensional, this implies the rank-nullity theorem:. Remarks I The kernel of a linear transformation is a. (The dimension of the image space is sometimes called the rank of T, and the dimension of the kernel is sometimes called the nullity of T. The range of A is the columns space of A. The linear transformation t 1 is the orthogonal reflection in the line y = x. Then (1) is a subspace of. SPECIFY THE VECTOR SPACES | {
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is the orthogonal reflection in the line y = x. Then (1) is a subspace of. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. The challenge is to find a transformation -> , such that the transformed dataset is linearly separable in. Recall: Linear Transformations De nition A transformation T : Rn!Rm is alinear transformationif it satis es the following two properties for all ~x;~y 2Rn and all (scalars) a 2R. 3 (Nullity). We have some fundamental concepts underlying linear transformations, such as the kernel and the image of a linear transformation, which are analogous to the zeros and range of a function. the kernel of a a linear transformation is the set of vectors in the null space of the matrix for that linear transformation. 4 Linear Transformations The operations \+" and \" provide a linear structure on vector space V. Analysis & Implementation Details. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. Conversely any linear fractional transformation is a composition of simple trans-formations. But, if we apply transformation X² to get: New Feature: X = np. If T(~x) = A~x, then the kernel of T is also called the kernel of A. Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Sx— 15y +4z x. Support vector machine with a polynomial kernel can generate a non-linear decision boundary using those polynomial features. The disadvantages are: 1) If the data is linearly separable in the expanded feature space, the linear SVM maximizes the margin better and can lead to a sparser solution. 1 LINEAR TRANSFORMATIONS 217 so that T is a linear transformation. Then, which of the following statements is always true?. The nullspace of a linear operator A is N(A) = {x ∈ X: Ax = 0}. Preimage and kernel | {
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is always true?. The nullspace of a linear operator A is N(A) = {x ∈ X: Ax = 0}. Preimage and kernel example. The range of L is the set of all vectors b ∈ W such that the equation L(x) = b has a solution. Summary: Kernel 1. Define the transformation \Omega: L(V,W) \to M_{m \times n} (\mathbb{R}) Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Gaussian Kernel always provides a value between 0 and 1. We write ker(A) or ker(T). However, there is also a limited amount of support for working with sparse matrices and vectors. im (T): Image of a transformation. on the order of 1000 or less since the algorithm is cubic in the number of features. And if the transformation is equal to some matrix times some vector, and we know that any linear transformation can be written as a matrix vector product, then the kernel of T is the same thing as the null space of A. Note that the range of the linear transformation T is the same as the range of the matrix A.$$ Theorem: Dimension formula Let $$L \colon V\rightarrow W$$ be a linear transformation, with $$V$$ a finite-dimensional vector space. N(T) is also referred to as the kernel of T. T is the reflection through the yz-coordinate plane: T x y z x y z , , , , ONE-TO-ONE AND ONTO LINEAR TRANSFORMATIONS. Let T: R 3!R3 be the transformation on R which re ects every vector across the plane x+y+z= 0. Find the kernel of the linear transformation. How to find the kernel of a linear transformation? Let B∈V =Mn(K) and let CB :V →V be the map defined by CB(A)=AB−BA. Then the kernel of L is de ned to be: ker(L) := fv 2V : L(v) = ~0g V i. The kernel of A are all solutions to the linear system Ax = 0. Question: Why is a linear transformation called “linear”?. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online | {
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network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This MATLAB function returns predicted class labels for each observation in the predictor data X based on the binary Gaussian kernel classification model Mdl. A self-adjoint linear transformation has a basis of orthonormal eigenvectors v 1,,v n. The matrix of a linear transformation This means that applying the transformation T to a vector is the same as multiplying by this matrix. It doesn't hurt to have it, but it isn't necessary here (in finding the kernel). This makes it possible to "turn around" all the arrows to create the inverse linear transformation $\ltinverse{T}$. Thus, the kernel consists of all matrices of the form [a b] [0 a] for a, b ∈ K; hence the nullity = 2. nan_euclidean_distances (X) Calculate the euclidean distances in the presence of missing values. Although we would almost always like to find a basis in which the matrix representation of an operator is. KPCA with linear kernel is the same as standard PCA. Then the Kernel of the linear transformation T is all elements of the vector space V that get mapped onto the zero element of the vector space W. If T isn't an isomorphism find bases of the kernel and image of T, and. Transformation Matrices. Now, let $\phi: V\longrightarrow W$ be a linear mapping/transformation between the two vector spaces. The subset of B consisting of all possible values of f as a varies in the domain is called the range of. Construct a linear transformation f and vector Y so that the system takes the form f(X)=Y. The next theorem is the key result of this chapter. The following examples illustrate the syntax. Two examples of linear transformations T :R2 → R2 are rotations around the origin and reflections along a line through the origin. Definition of the Image of linear map 𝐋. large values of , and clearly approach the linear regression; the | {
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of the Image of linear map 𝐋. large values of , and clearly approach the linear regression; the curves shown in red are for smaller values of. Polynomial Kernel. Note that the range of the linear transformation T is the same as the range of the matrix A. Find a basis for the kernel of T and the range of T. Let’s begin by rst nding the image and kernel of a linear transformation. This paper studies the conditions for the idempotent transformation and the idempotent rank transformation direct sum decomposition for finite dimension of linear space. The following examples illustrate the syntax. Now is the time to redefine your true self using Slader’s free Linear Algebra: A Modern Introduction answers. Definition Kernel and Image. These functions can be different types. To connect linear algebra to other fields both within and without mathematics. KPCA with linear kernel is the same as standard PCA. This mapping is called the orthogonal projection of V onto W. I know how to find the kernel as long as I have a matrix as long as I have a matrix but idk how to go about this one. (d)The rank of a linear transformation equals the dimension of its kernel. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. (b)Find a linear transformation whose kernel is Sand whose range is S?. Given two vector spaces V and W and a linear transformation L : V !W we de ne a set: Ker(L) = f~v 2V jL(~v) = ~0g= L 1(f~0g) which we call the kernel of L. There are some important concepts students must master to solve linear transformation problems, such as kernel, image, nullity, and rank of a linear transformation. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. In this paper, we study metric learning as a | {
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transformation between {eq}V_1 {/eq} and {eq}V_2. In this paper, we study metric learning as a problem of learning a linear transformation of the input data. Problem: I can't find answer to a problem. (e)The nullity of a linear transformation equals the dimension of its range. Since the nullity is the dimension of the null space, we see that the nullity of T is 0 since the dimension of the zero vector space is 0. 17 The rank of a linear map is less than or equal to the dimension of the domain. IV Image and Kernel of Linear Transformations Motivation: In the last class, we looked at the linearity of this function: F : R3 R2 F(x,y,z)=(x+y+z,2x-3y+4z) How does a 3 dimensional space get ‘mapped into’ a 2 dimensional space? At least one dimension ‘collapses’, or disappears. 2 Kernel and Range of linear Transfor-mation We will essentially, skip this section. (a) Find a basis of the range of P. Describe the kernel and range of a linear transformation. The image of a linear transformation or matrix is the span of the vectors of the linear transformation. Morphological transformations are some simple operations based on the image shape. (c) Find the nullity and rank of P. sage : M = MatrixSpace ( IntegerRing (), 4 , 2 )( range ( 8 )) sage : M. Then, ker(L) is a subspace of V. We will start with Hinge Loss and see how the optimization/cost function can be changed to use the Kernel Function,. If M is singular there must be a linear combination of rows of M that sums to the zero row vector. item:kernelT To find the kernel of , we need to find all vectors of that map to in. 2 The kernel and range of a linear transformation. The set consisting of all the vectors v 2V such that T(v) = 0 is called the kernel of T. This can be defined set-theoretically as follows:. MATH 316U (003) - 10. S: ℝ3 → ℝ3. Therefore, w 1 and w 2 form an orthonormal basis of the kernel of A. If T(u) = u x v find the kernel and range of the transformation as well as the matrix for the transformation if v = i | {
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the kernel and range of the transformation as well as the matrix for the transformation if v = i (which I am assuming is (1,0,0)). to construct the whitening transformation matrix for orthogonalizing the linear subspaces in the feature space F. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. Before we do that, let us give a few definitions. 0000 Today we are going to continue our discussion of the kernel and range of a linear map of a linear transformation. Finding the kernel of a linear transformation involving an integral. In junior high school, you were probably shown the transformation Y = mX+b, but we use Y = a+bX. range(T)={A in W | there exists B in V such that T(B)=A}. Algebra Examples. {\mathbb R}^m. Other Kernel Methods •A lesson learned in SVM: a linear algorithm in the feature space is equivalent to a non-linear algorithm in the input space •Classic linear algorithms can be generalized to its non-linear version by going to the feature space –Kernel principal component analysis, kernel independent component analysis, kernel. Definition 6. Some linear transformations possess one, or both, of two key properties, which go by the names injective and surjective. transformation, the kernel and the image. Find more Mathematics widgets in Wolfram|Alpha. Note: It is convention to use the Greek letter 'phi' for this transformation , so I'll use. Lesson: Image and Kernel of Linear Transformation Mathematics. Determine whether T is an isomorphism. ∆ Let T: V ‘ W be a linear transformation, and let {eá} be a basis for V. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. We illustrated the quadratic kernel in quad-kernel. , the solutions of the equation A~x = ~ 0. Linear algebra - | {
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the quadratic kernel in quad-kernel. , the solutions of the equation A~x = ~ 0. Linear algebra - Practice problems for midterm 2 1. We have some fundamental concepts underlying linear transformations, such as the kernel and the image of a linear transformation, which are analogous to the zeros and range of a function. This paper studies the conditions for the idempotent transformation and the idempotent rank transformation direct sum decomposition for finite dimension of linear space. More importantly, as an injective linear transformation, the kernel is trivial (Theorem KILT), so each pre-image is a single vector. To take an easy example, suppose we have a linear transformation on R 2 that maps (x, y) to (4x+ 2y, 2x+ y). KERNEL AND RANGE OF LINEAR TRANSFORMATION199 6. The transformation is selected from a parametric family, which is allowed to be quite general in our theoretical study. For a linear transformation T from Rn to Rm, † im(T) is a subset of the codomain Rm of T, and † ker(T) is a subset of the domain Rn. Finding a basis of the null space of a matrix. Find the matrix of the given linear transformation T with respect to the given basis. Note that N(T) is a subspace of V, so its dimension can be de ned. We solve by finding the corresponding 2 x 3 matrix A, and find its null space and column span. Such a repre-sentation is frequently called a canonical form. Let V;W be vector spaces over a eld F. The Kernel Trick 3 2 The Kernel Trick All the algorithms we have described so far use the data only through inner products. manhattan_distances (X[, Y, …]) Compute the L1 distances between the vectors in X and Y. 0)( =vT ker( ) {v | (v) 0, v }T T V= = ∀ ∈. The kernel and image of a matrix A of T is defined as the kernel and image of T. If V is finite-dimensional, then so are Im(T) and ker(T), anddim(Im(T))+dim(ker(T))=dimV. Since the correlation coefficient is maximized when a scatter diagram is linear, we can use the same approach above to find the most normal | {
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when a scatter diagram is linear, we can use the same approach above to find the most normal transformation. Find the kernel of the linear transformation L: V→W. Determine whether the following functions are linear transformations. All Slader step-by-step solutions are FREE. IV Image and Kernel of Linear Transformations Motivation: In the last class, we looked at the linearity of this function: F : R3 R2 F(x,y,z)=(x+y+z,2x-3y+4z) How does a 3 dimensional space get ‘mapped into’ a 2 dimensional space? At least one dimension ‘collapses’, or disappears. However, there is also a limited amount of support for working with sparse matrices and vectors. We will start with Hinge Loss and see how the optimization/cost function can be changed to use the Kernel Function,. Then T is a linear transformation. {\mathbb R}^n. Discuss this quiz (Key: correct, incorrect, partially correct. Let’s begin by rst nding the image and kernel of a linear transformation. Find the matrix of the given linear transformation T with respect to the given basis. We prove the theorems relating to kernel and image of linear transformation. To connect linear algebra to other fields both within and without mathematics. The general solution is a linear combination of the elements of a basis for the kernel, with the coefficients being arbitrary constants. Gauss-Jordan elimination yields: Thus, the kernel of consists of all elements of the form:. Null space. A= [-3, -2 , 4. What is the outcome of solving the problem?. 1 Example Clearly, the data on the left in figure 1 is not linearly separable. If T(~x) = A~x, then the kernel of T is also called the kernel of A. How Linear Transformations Affect the Mean and Variance. Now is the time to redefine your true self using Slader’s free Linear Algebra: A Modern Introduction answers. Note that the squares of s add, not the s 's themselves. [Solution] To get an orthonormal basis of W, we use Gram-Schmidt process for v1 and v2. Step-by-Step Examples. The next example | {
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basis of W, we use Gram-Schmidt process for v1 and v2. Step-by-Step Examples. The next example illustrates how to find this matrix. De nition 1. Finding the kernel of the linear transformation. Then the Kernel of the linear transformation T is all elements of the vector space V that get mapped onto the zero element of the vector space W. The only solution is x = y = 0, and thus the zero vector (0. (2) is injective if and only if. Of course we can. transformation, the kernel and the image. It is normally performed on binary images. Preimage and kernel example. The kernel of a linear operator is the set of solutions to T(u) = 0, and the range is all vectors in W which can be expressed as T(u) for some u 2V. SVG image not dispayed. Hello and welcome back to Educator. If a linear transformation T: R n → R m has an inverse function, then m = n. To find the null space we must first reduce the #3xx3# matrix found above to row echelon form. Let us say I have 3 vectors in v that map to 0, those three vectors, that is my kernel of my linear map. The null space of T, denoted N(T), is de ned as N(T) := fv2V: T(v) = 0g: Remark 3. Find the matrix of the orthogonal projection onto W. (b) Find the matrix representation of L with respect to the standard basis 1;x;x2. 1 Matrix Linear Transformations Every m nmatrix Aover Fde nes linear transformationT A: Fn!Fmvia matrix multiplication. Create a system of equations from the vector equation. [Solution] To get an orthonormal basis of W, we use Gram-Schmidt process for v1 and v2. The kernel of L is the solution set of the homogeneous linear equation L(x) = 0. Basically, the kernel of a linear map, from a vector space v to a vector space w is all those vectors in v that map to the 0 vector. Find the kernel of the linear transformation L: V→W. visualize what the particular transformation is doing. The range of L is the set of all vectors b ∈ W such that the equation L(x) = b has a solution. , it can be applied to unseen data. The linear | {
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b ∈ W such that the equation L(x) = b has a solution. , it can be applied to unseen data. The linear transformation t 2 is the orthogonal projection on the x-axis. 2 The kernel and range of a linear transformation. TRUE To show this we show it is a subspace Col A is the set of a vectors that can be written as Ax for some x. (c)The range of a linear transformation is a subspace of the co-domain. SUBSCRIBE to the channel and. To find the kernel, you just need to determine the dimensionality of the solution space to the linear system. The Gaussian is a self-similar function. This mapping is called the orthogonal projection of V onto W. The spline can also be used for prediction. The following charts show some of the ideas of non-linear transformation. Now, let $\phi: V\longrightarrow W$ be a linear mapping/transformation between the two vector spaces. Justify your answers. The disadvantages are: 1) If the data is linearly separable in the expanded feature space, the linear SVM maximizes the margin better and can lead to a sparser solution. To compute the kernel, find the null space of the matrix of the linear transformation, which is the same to find. Please wait until "Ready!" is written in the 1,1 entry of the spreadsheet. Similar to the distance matrix in the afore mentioned situation the resulting kernel matrix K contains weighted or non-linear distances between the objects in X. Then for any x ∞ V we have x = Íxáeá, and hence T(x) = T(Íxáeá) = ÍxáT(eá). It is the set of vectors, the collection of vectors that end up under the transformation mapping to 0. Find the rank and nullity of a linear transformation from R^3 to R^2. {\mathbb R}^m. There are some important concepts students must master to solve linear transformation problems, such as kernel, image, nullity, and rank of a linear transformation. The Kernel of a Linear Transformation: Suppose that {eq}V_1 {/eq} and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation | {
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and {eq}V_2 {/eq} are two vector spaces, and {eq}T:V_1 \to V_2 {/eq} is a linear transformation between {eq}V_1 {/eq} and {eq}V_2. If there are page buffers, the total number of bytes in the page buffer area is 'data_len'. TThis quiz is designed to test your knowledge of linear transformations and related concepts such as rank, nullity, invertibility, null space, range, etc. SKBs are composed of a linear data buffer, and optionally a set of 1 or more page buffers. Define pre-image of U, denoted T -1. If T(~x) = A~x, then the kernel of T is also called the kernel of A. Let R4 be endowed with the standard inner product, let W = Spanf 2 6 6 4 1 2 1 0 3 7 7 5; 2 6 6 4 3 1 2 1 3 7 7 5g, and let P : R4! R4 be the orthogonal projection in R4 onto W. Find polynomial(s) p i(t) that span the kernel of T. The idea of a linear transformation is that one variable is mapped onto another in a 1-to-1 fashion. Theorem Let T:V→W be a linear transformation. {\mathbb R}^n. Course goals. Specifically, if T: n m is a linear transformation, then there is a unique m n matrix, A, such that T x Ax for all x n. We define the kernel of $\phi$ to be. Vector Spaces and Linear Transformations Beifang Chen Fall 2006 1 Vector spaces A vector space is a nonempty set V, whose objects are called vectors, equipped with two operations, called addition and scalar multiplication: For any two vectors u, v in V and a scalar c, there are unique vectors u+v and cu in V such that the following properties are satisfled. One thing to look out for are the tails of the distribution vs. – Suppose we have a linear transformation f: Rn!. For example linear, nonlinear, polynomial, radial basis function. {\mathbb R}^m. It is the set of vectors, the collection of vectors that end up under the transformation mapping to 0. What is the range of T in R2?. It doesn't hurt to have it, but it isn't necessary here (in finding the kernel). De nition 3. One-to-One linear transformations: In college algebra, we could perform a | {
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the kernel). De nition 3. One-to-One linear transformations: In college algebra, we could perform a horizontal line test to determine if a function was one-to-one, i. The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation). Demonstrate: A mapping between two sets L: V !W. KERNEL AND RANGE OF LINEAR TRANSFORMATION199 6. Before we look at some examples of the null spaces of linear transformations, we will first establish that the null space can never be equal to the empty set, in. We describe the range by giving its basis. First here is a definition of what is meant by the image and kernel of a linear transformation. ) T: R^2 rightarrow R^2, T(x, y) = (x + 2y, y - x) ker(T) = {: x, y R} T(v) = Av represents the linear transformation T. Remarks I The kernel of a linear transformation is a. linear transformation. Step-by-Step Examples. TRUE Remember that Ax gives a linear combination of columns of A using x entries as weights. Note that the squares of s add, not the s 's themselves. Synonyms: kernel onto A linear transformation, T, is onto if its range is all of its codomain, not merely a subspace. More Examples of Linear Transformations: solutions: 6: More on Bases of $$\mathbb{R}^n$$, Matrix Products: solutions: 7: Matrix Inverses: solutions: 8: Coordinates: solutions: 9: Image and Kernel of a Linear Transformation, Introduction to Linear Independence: solutions: 10: Subspaces of $$\mathbb{R}^n$$, Bases and Linear Independence. Find the matrix of the given linear transformation T with respect to the given basis. There are some important concepts students must master to solve linear transformation problems, such as kernel, image, nullity, and rank of a linear transformation. Since Whas dimension n, the image of Tmust equal W. (c)Find a linear transformation whose kernel is S?and whose range is S. [Linear Algebra] Finding the kernel of a linear transformation. Similarly, we say a linear | {
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is S. [Linear Algebra] Finding the kernel of a linear transformation. Similarly, we say a linear transformation T: max 1 i n di +2 r 2R2 n (p 2+ln r 1 )) 1 n+1 where the support of the distribution D is assumed to be contained in a ball of radius R. Explainer +9; Read. The following section goes through the the different objective functions and shows how to use Kernel Tricks for Non Linear SVM. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It is an extension of Principal Component Analysis (PCA) - which is a linear dimensionality reduction technique - using kernel methods. You can even pass in a custom kernel. im (T): Image of a transformation. T [x, y, z, w] = [x + 2y + z - w] [2x + 3y - z + w] LHS of equation is a 4 row by 1 column matrix. It is given by the inner product plus an optional constant c. From this, it follows that the image of L is isomorphic to the quotient of V by the kernel: ≅ / (). (b) The dual space V ∗ of the vector space V is the set of all linear functionals on V. To find the kernel of a matrix A is the same as to solve the system AX = 0, and one usually does this by putting A in rref. Griti is a learning community for students by students. That is it. 3, -3 , 1] Find the basis of the image of a linear transformation T defined by T(x)=Ax. , Mladenov, M. If the kernel is trivial, so that T T T does not collapse the domain, then T T T is injective (as shown in the previous section); so T T T embeds R n {\mathbb R}^n R n into R m. We have step-by-step solutions for your textbooks written by Bartleby experts! Find the nullity of the linear transformation T : M n n → ℝ defined by T ( A ) = tr ( A ). De nition 1. A = [2 1] [3 4]. The kernel of a linear transformation is a vector space. Suppose T:R^3 \\to R^3,\\quad T(x,y,z) = (x + 2y, y + 2z, z + 2x) Part of Solution: The problem is solved like | {
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\\to R^3,\\quad T(x,y,z) = (x + 2y, y + 2z, z + 2x) Part of Solution: The problem is solved like this: A =. Solving systems of nonlinear equa- tions can be tricky. ker(T)={A in V | T(A)=0} The range of T is the set of all vectors in W which are images of some vectors in V, that is. Handbook ofNEURAL NETWORK SIGNAL PROCESSING© 2002 by CRC Press LLC THE ELECTRICAL ENGINEERING AND APPLIED SIGNAL P. Let P n(x) be the space of polynomials in x of degree less than or equal to n, and consider the derivative operator d dx. Let $$T:V\rightarrow W$$ be a linear transformation where $$V$$ and $$W$$ be vector spaces with scalars coming from the same field $$\mathbb{F}$$. Similarly, we say a linear transformation T: max 1 i n di +2 r 2R2 n (p 2+ln r 1 )) 1 n+1 where the support of the distribution D is assumed to be contained in a ball of radius R. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The kernel of L is the solution set of the homogeneous linear equation L(x) = 0. In fact, 4x+ 2y= 2(2x+ y) so those are the same equation which is equivalent to y= -2x. (b)The kernel of a linear transformation is a subspace of the domain. Trying to use matrices and matrix methods is almost a waste of time in this problem. Choose a simple yet non-trivial linear transformation with a non-trivial kernel and verify the above claim for the transformation you choose. The range of A is the columns space of A. The algorithm: The idea behind kernelml is simple. The most common form of radial basis function is a Gaussian distribution, calculated as:. visualize what the particular transformation is doing. Define pre-image of U, denoted T -1. If T(~x) = A~x, then the kernel of T is also called the kernel of A. Let be a linear transformation. Find the rank and nullity of a linear transformation from R^3 to R^2. 4 Linear Transformations The operations \+" | {
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and nullity of a linear transformation from R^3 to R^2. 4 Linear Transformations The operations \+" and \" provide a linear structure on vector space V. F respects linear combinations, + q [F (VIC) of the following hold: i. Question: Why is a linear transformation called “linear”?. suppose T(x,y,z) = ( 2x-3y, x+4y-z, -x-7y+5z ) be a linear transformation. Similarly, we say a linear transformation T: max 1 i n di +2 r 2R2 n (p 2+ln r 1 )) 1 n+1 where the support of the distribution D is assumed to be contained in a ball of radius R. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. The dimension of the kernel of T is the same as the dimension of its null space and is called the nullity of the transformation. Find more Mathematics widgets in Wolfram|Alpha. Let $$T:V\rightarrow W$$ be a linear transformation where $$V$$ and $$W$$ be vector spaces with scalars coming from the same field $$\mathbb{F}$$. 2) When there is a large dataset linear SVM takes lesser time to train and predict compared to a Kernelized SVM in the expanded feature space. We solve by finding the corresponding 2 x 3 matrix A, and find its null space and column span. [Solution] To get an orthonormal basis of W, we use Gram-Schmidt process for v1 and v2. Similarly, a vector v is in the kernel of a linear transformation T if and only if T(v)=0. Griti is a learning community for students by students. (If all real numbers are solutions, enter REALS. $$Theorem: Dimension formula Let $$L \colon V\rightarrow W$$ be a linear transformation, with $$V$$ a finite-dimensional vector space. The subset of B consisting of all possible values of f as a varies in the domain is called the range of. Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Sx— 15y +4z x. , the solutions of the equation A~x = ~0. The | {
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Find a basis Find a basis Find a basis Sx— 15y +4z x. , the solutions of the equation A~x = ~0. The spline can also be used for prediction. Find the kernel of the linear transformation L: V→W. The Polynomial kernel is a non-stationary kernel. The Matrix of a Linear Transformation We have seen that any matrix transformation x Ax is a linear transformation. These functions can be different types. Most off-the-shelf classifiers allow the user to specify one of three popular kernels: the polynomial, radial basis function, and sigmoid kernel. Null space. Griti is a learning community for students by students. Define the transformation \Omega: L(V,W) \to M_{m \times n} (\mathbb{R}) Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Let L : V →W be a linear transformation. Demonstrate: A mapping between two sets L: V !W. 2 (The Kernel and Range)/3. For example: random forests theoretically use feature selection but effectively may not, support vector machines use L2 regularization etc. The kernel of T, denoted by ker(T), is the set of all vectors x in Rn such that T(x) = Ax = 0. Q2 The Dimension of The Image and Kernel of a Linear Transformation 50 Points Q2. Determine whether the following functions are linear transformations. Before we look at some examples of the null spaces of linear transformations, we will first establish that the null space can never be equal to the empty set, in. Discuss this quiz (Key: correct, incorrect, partially correct. Then (1) is a subspace of. To help the students develop the ability to solve problems using linear algebra. Create a system of equations from the vector equation. Use the parameter update history in a machine learning model to decide how to update the next parameter set. Because Tis one-to-one, the dimension of the image of Tmust be n. You should think about | {
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set. Because Tis one-to-one, the dimension of the image of Tmust be n. You should think about something called the null space. Define by Observe that. We describe the range by giving its basis. The kernel of T, also called the null space of T, is the inverse image of the zero vector, 0, of W, ker(T) = T 1(0) = fv 2VjTv = 0g: It's sometimes denoted N(T) for null space of T. SPECIFY THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. We define the kernel of $\phi$ to be. Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Find a basis Sx— 15y +4z x. THE PROPERTIES OF DETERMINANTS a. Recall that if a set of vectors v 1;v 2;:::;v n is linearly independent, that means that the linear combination c. The kernel of a linear transformation {eq}L: V\rightarrow V {/eq} is the set of all polynomials such that {eq}L(p(t))=0 {/eq} Here, {eq}p(t) {/eq} is a polynomial. Image of a subset under a transformation. A linear map L∶V → W is called a linear isomorphism if ker(L) = 0 and L(V) = W. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. It is one-one if its kernel is just the zero vector, and it is. 6, -1 ,-3-3 , 3 ,-1. Discuss this quiz (Key: correct, incorrect, partially correct. RHS of equation is a 2 row by 1 column matrix. We build thousands of video walkthroughs for your college courses taught by student experts who got an A+. Theorem If the linear equation L(x) = b is solvable then the. Hello and welcome back to Educator. The nullspace of a linear operator A is N(A) = {x ∈ X: Ax = 0}. We write ker(A) or ker(T). And we saw that earlier in the video. Let A = 2 4 0. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. (7 pt total) Linear Transformations. | {
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corresponding homogeneous linear equations, AX = 0 or BX = 0. (7 pt total) Linear Transformations. Shed the societal and cultural narratives holding you back and let free step-by-step Linear Algebra: A Modern Introduction textbook solutions reorient your old paradigms. 1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Transformation Matrices. (If all real numbers are solutions, enter REALS. There entires in these lists are arguable. Preimage and kernel example. 2 The Kernel and Range of a Linear Transformation4. More on matrix addition and scalar multiplication. as in Definition 1. Namely, linear transformation matrix learned in the high dimensional feature space can more appropriately map samples into their class labels and has more powerful discriminating ability. Next, we find the range of T. Use the kernel and image to determine if a linear transformation is one to one or onto. T [x, y, z, w] = [x + 2y + z - w] [2x + 3y - z + w] LHS of equation is a 4 row by 1 column matrix. A linear transformation T: R n → R m has an inverse function if and only if its kernel contains just the zero vector and its range is its whole codomain. nan_euclidean_distances (X) Calculate the euclidean distances in the presence of missing values. Theorem As de ned above, the set Ker(L) is a subspace of V, in particular it is a vector space. Use automated training to quickly try a selection of model types, and then explore promising models interactively. For instance, sklearn's SVM implementation svm. One can row reduce A to the identity matrix. We build thousands of video walkthroughs for your college courses taught by student experts who got an A+. Affine transformations", you can find examples of the use of linear transformations, which can be defined as a mapping between two vector spaces that preserves linearity. The problem comes when finding the Kernel basis. Justify your answers. These solutions are not necessarily a vector | {
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when finding the Kernel basis. Justify your answers. These solutions are not necessarily a vector space. ] all keywords, in any order at least one, that exact phrase parts of words whole words. To see why image relates to a linear transformation and a matrix, see the article on linear. The $$\textit{nullity}$$ of a linear transformation is the dimension of the kernel, written$$ nul L=\dim \ker L. Ker(T) is the solution space to [T]x= 0. [Linear Algebra] Finding the kernel of a linear transformation. ANSWER Let p = ax2 +bx +c. Thus matrix multiplication provides a wealth of examples of linear transformations between real vector spaces. Anyway, hopefully you found that reasonably. (b)Find a linear transformation whose kernel is Sand whose range is S?. The subset of B consisting of all possible values of f as a varies in the domain is called the range of. 2 The kernel and range of a linear transformation. This theorem implies that every linear transformation is also a matrix transformation. Let P n(x) be the space of polynomials in x of degree less than or equal to n, and consider the derivative operator d dx. S: ℝ3 → ℝ3. com and welcome back to linear algebra. manhattan_distances (X[, Y, …]) Compute the L1 distances between the vectors in X and Y. The image of a linear transformation contains 0 and is closed under addition and scalar multiplication. Let be a linear transformation. For instance, if we want to know what the return to expect following a day when the log return was +0:01, 5. (some people call this the nullspace of L). We will see in the next subsection that the opposite is true: every linear transformation is a matrix transformation; we just haven't computed its matrix yet. Gaussian Kernel always provides a value between 0 and 1. Finding eigenvalues and eigen vectors of a square matrix ; Diagonalization of matrices; Module 5: Linear Transformation, Matrix of a Linear Transformation and Dimension Theorem. Similarly, we say a linear transformation T: mthen | {
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of a Linear Transformation and Dimension Theorem. Similarly, we say a linear transformation T: mthen there exists infinite solutions. AND LINEAR TRANSFORMATIONS Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices are Find a polynomial p in P2 that spans the kernel of T, and describe the range of T. Then for any x ∞ V we have x = Íxáeá, and hence T(x) = T(Íxáeá) = ÍxáT(eá). 0004 From the previous lesson, we left it off defining what the range of a linear map is. The equationof-state formulation is based on the monotoric strain-hardening rule app1ied to the primarymore ». In this section, you will learn most commonly used non-linear regression and how to transform them into linear regression. Consider the linear system x+2y +3z = 1 2xy +2z =9 1. Note: Because Rn is a "larger" set than Rm when m < n, it should not be possible to map Rn to Rm in a one-to-one fashion. can be impractical to use. 1 2 -3 : 1/ 5 y 1 0 0 0 : - 7/. THE KERNEL IS A SUBSPACE: Let L : V !W be a linear transformation. Support vector machine with a polynomial kernel can generate a non-linear decision boundary using those polynomial features. Thus, the kernel of a matrix transformation T(x)=Ax is the null space of A. | {
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# Polynomial Division Problem
Polynomial Division Problem
Calculate the following and state the remainder:
2x^3+x^2-22x+20 divided by 2x-3
Thanks!
#### topsquark
Math Team
Polynomial Division Problem
Calculate the following and state the remainder:
2x^3+x^2-22x+20 divided by 2x-3
Thanks!
How are you to do this? You could use long division or sythnetic division.
What have you been able to work out?
-Dan
I calculated using long division and got the following:
x^2+2x-8 remainder -4
This question is from textbook and solution used synthetic division, but answer is different than mine...
Textbook solution gives: x^2+2x-8 remainder -2
#### skeeter
Math Team
$\dfrac{2x^3 +x^2-22x+20}{2x-3} = \dfrac{x^3 + \frac{x^2}{2} - 11x + 10}{x-\frac{3}{2}}$
synthetic division for the right side ...
Code:
[3/2] 1 1/2 -11 10
3/2 3 -12
------------------------------
1 2 -8 | -2
$\dfrac{x^3 + \frac{x^2}{2} - 11x + 10}{x-\frac{3}{2}} = x^2+2x-8 - \dfrac{2}{x-\frac{3}{2}} = x^2 + 2x - 8 - \dfrac{4}{2x-3}$
2 people
@ skeeter
Thanks, I see that (-2)/(x-3/2) is equivalent to (-4)/(2x-3)
So is the textbook incorrect? Should the remainder be (-2)/(x-3/2) instead of just -2?
Also, how to solve using long division then?
Here's txt solution using synthetic division:
https://imgur.com/BF0ZgTH
Last edited:
#### skeeter
Math Team
Code:
x^2 + 2x - 8
--------------------------
2x - 3 | 2x^3 + x^2 - 22x + 20
2x^3 - 3x^2
--------------------
4x^2 - 22x + 20
4x^2 - 6x
---------------
-16x + 20
-16x + 24
----------
-4
according to the remainder theorem, the remainder is -4, because ...
$f(x) = 2x^3 + x^2 - 22x + 20 \implies f\left(\frac{3}{2}\right) = -4$
Last edited:
2 people
@ Skeeter
Okay, so textbook is incorrect. Thanks for clarifying!
:-D
But I still don't understand how synthetic division gave incorrect remainder? Doesn't the -2 at the end of your synthetic division represent the remainder?
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Last edited:
#### skeeter
Math Team
-2 is the remainder for $g(x) = x^2 + \dfrac{x^2}{2} - 11x + 10 \, \text{ since }\, g\left(\frac{3}{2}\right) = -2$
$2 \cdot g(x) = f(x) \implies$ the remainder for g(x) is half that for f(x).
It's all a matter of which polynomial is used to determine the remainder ... I support -4 as the remainder since f(x) divided by (2x-3) was the original problem. | {
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# difference between quotient rule and product rule
Product rule :
$$\frac{d}{dx} \big(f(x)\cdot g(x)\big)=f'(x)\cdot g(x)+f(x)\cdot g' (x)$$
Quotient rule :
$$\frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$$
Suppose, the following is given in question. $$y=\frac{2x^3+4x^2+2}{3x^2+2x^3}$$
Simply, this is looking like Quotient rule. But, if I follow arrange the equation following way
$$y=(2x^3+4x^2+2)(3x^2+2x^3)^{-1}$$
Then, we can solve it using Product rule. As I was solving earlier problems in a pdf book using Product rule. I think both answers are correct. But, my question is, How does a Physicist and Mathematician solve this type question? Even, is it OK to use Product rule instead of Quotient rule in University and Real Life?
• Your just deriving the quotient rule on the fly, rather than assuming it is true and then applying it directly; there is nothing wrong with doing the former.
– user785085
May 8, 2021 at 13:39
• First thing a mathematician would do is write $$y=1+\frac{x^2+2}{2x^3+3x^2}$$ before taking the derivative. May 8, 2021 at 16:14
• What do you mean "I think both answers are correct."? If you do the math properly, you'll get the same answer with both methods. May 9, 2021 at 0:10
• Real physicists and mathematicians stick the whole thing into their computer algebra system and don't worry about the precise algorithms it follows, unless they have a particular reason to risk silly errors by doing that kind of calculation by hand. Symbolic differentiation is a solved problem; one doesn't earn any "purity points" in the real world by doing it the hard way. May 9, 2021 at 0:33
• @JosephSible-ReinstateMonica Yes! I got same answer :)
– user876873
May 9, 2021 at 4:30 | {
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Note that $$((g(x))^{-1})'=-g'(x)(g(x))^{-2}$$. Then, applying the product rule: $$\left(\frac{f(x)}{g(x)}\right)'=\left(f(x)\cdot \frac{1}{g(x)}\right)'=\frac{f'(x)}{g(x)}+\frac{-g'(x)f(x)}{(g(x))^2}=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}$$ which is the quotient rule
• That's very helpful. But, it was my main question How does a Physicist and Mathematician solve this type question? Even, is it OK to use Product rule instead of Quotient rule in University and Real Life?
– user876873
May 8, 2021 at 13:44
• @Istiak Please don't keep making that bold. Sure, it's OK, because any valid proof is OK; there are no "rules" beyond that. To be honest, I hardly ever see mathematicians or physicists explicitly using the quotient rule. It seems to be more for teaching than anything else.
– J.G.
May 8, 2021 at 13:45
• I'd always use the quotient rule on a quotient because it is usually much simpler to work with $g'$ than $(\frac1g)'$ May 8, 2021 at 13:45
• @J.G. Actually, I had read in a meta post that is saying,you should bold when it may need to be attracted or, something just like this. I don't remember. That's why I was just making that bold
– user876873
May 8, 2021 at 13:48
• @Istiak I see, cool. Thanks, though, for your edit that means you now only bold it in the question.
– J.G.
May 8, 2021 at 13:52
How does a Physicist and Mathematician solve this type question? Even, is it OK to use Product rule instead of Quotient rule in University and Real Life?
is that any experienced scientist knows several methods to solve problems and uses those that are most convenient for them at that particular time.
I would look at that derivative and use the quotient rule. But if there was something in the source of the problem that suggested that it made more sense to write the denominator as $$(3x^2+2x^3)^{-1}$$ then the product rule would be more appropriate.
the quotient rule is not a separate and non-compatible rule. It is just the product rule inserting $$\frac 1 g$$ instead of $$g$$. | {
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let's see...
to make it clear... I show you, prove it for you, how quotient rule is just compatible with product law.
$$\left(\frac f g\right)'=\left(f\cdot \frac 1 g\right)'$$
as product law says:
$$=f' \cdot \frac 1 g +f\cdot \left (\frac 1 g\right)'$$
while $$\displaystyle\left(\frac 1 g \right)' = \frac {-g'}{g^2}$$, we would have
$$=f'\cdot \frac 1 g+f\cdot \frac{-g'}{g^2}=\frac{f'\cdot g-f\cdot g'}{g^2}.$$
Note:
You may know that $$\displaystyle\left(\frac 1 h \right)' = \frac {-h'}{h^2}$$ could be calculated by product rule, as if one consider the product $$\displaystyle\left(\frac 1 h \cdot h \right) = 1$$, and calculate the derivative of both sides of the equation. one the left hand side we have a constant which may already know the derivative is $$0$$, but on the other side we see a product so by applying the product rule we have $$0=h' \cdot \frac 1 h + h\cdot \left (\frac 1 h \right)'.$$
Therefore $$h\cdot \left (\frac 1 h \right)'=-h' \cdot \frac 1 h.$$
And so $$\left (\frac 1 h \right)'=-h' \cdot \frac 1 {h^2} = -\frac {h'} {h^2}.$$
• $g^{-1}$ is a poor choice of notation here, as it usually means the inverse of $g$, and not the reciprocal. May 9, 2021 at 0:19
• @XanderHenderson I fixed the notation, thanks... as you remember I never used the $g^{-1}$ notation in the followings May 6 at 15:12 | {
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# What is wrong with the argument $1 = \lim_{n\to \infty} n/n = \lim_{n\to\infty} (1/n+1/n+\dotsb+1/n) = 0$?
Let we have
$$\begin{equation*} n\times\frac{1}{n}=\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}\mbox{ (n times)}. \end{equation*}$$
Taking $$\lim_{n\to\infty}$$ to both sides, we get
$$\begin{eqnarray*} \lim_{n\to\infty}1 &=& \lim_{n\to\infty}\left(\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}\right)\\ \Longrightarrow1&=&0+0+\cdots+0\mbox{ (n times)}\\ &=&0. \end{eqnarray*}$$
I am not an expert of math, and confused that where the confusion is.
• An informal answer here: The problem is the "dot...dot...dot" part. There is such thing as an indeterminate form, being zero times infinity. And that is typically NOT zero. With the "dot...dot...dot" notation, you are "reading" the problem as a finite amount of $1/n$ terms (even though it is not your intention) and hence you come up with zero as the (incorrect) answer. – imranfat Aug 5 at 2:17
• Also, I up-voted the question because it is a good question. You pose this question to introductory calculus students and I bet half of them cannot point out the issue. – imranfat Aug 5 at 2:19
• By your logic, on the left you also have $\lim n \cdot \frac{1}{n} = \lim_{n\to\infty} n \cdot \lim_{n\to\infty} \frac{1}{n} = \infty \cdot 0.$ – mjw Aug 5 at 2:20
• The problem is that you can't only take limit on $n$ only on part of the expression. There is the "$n$ times" which also depended on the $n$ you are taking to $\infty$. And no, you can't split the limits up in two in general. – user10354138 Aug 5 at 2:21
• – Angina Seng Aug 5 at 2:27
This is an excellent argument that we cannot in general find a limit by taking the limits of the parts of an expression.
When many students are first introduced to limit laws, they see their instructor go through a lot of complicated math in order to prove things that feel obvious. In this case, the relevant one is the addition law: | {
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$$\lim_{x \to c}\left(f(x) + g(x)\right) = \lim_{x\to c}f(x) + \lim_{x\to c}g(x)$$
This seems obvious, right? A limit means "what number does this expression get close to". Of course $$f(x) + g(x)$$ would get close to the sum of whatever $$f(x)$$ gets close to and whatever $$g(x)$$ gets close to. So why does the instructor (or the textbook) spend half a page messing around with $$\epsilon$$s and $$\delta$$s to prove the law?
The answer is because of exactly the sort of thing you've pointed out. There are situations where the "intuitive" approach to limits stops working, essentially because infinity is hard. For those situations, we need to rely on the proof. Crucially, in this case, the proof relies on there being only two things added together. This means, if we want to adhere perfectly to the law as stated, we have to jump through hoops like this:
\begin{align*} \lim_{x \to c} \left(f(x) + g(x) + h(x)\right) &= \lim_{x \to c} \left(\left(f(x) + g(x)\right) + h(x)\right)\\ &= \lim_{x \to c}\left(f(x) + g(x)\right) + \lim_{x \to c}h(x)\\ &= \lim_{x \to c}f(x) + \lim_{x \to c}g(x) + \lim_{x \to c}h(x) \end{align*}
We can do the same to deal with four, or five, or five hundred things added together. But how would we deal with $$n$$ things added together, when $$n$$ changes over the course of the limit? If we "peel off" one like I did above, there'd still be infinitely many left over. In other words, even with aggressive uses of this limit law, we can only handle sums of fixed size. One that "grows", like $$\frac1n + \frac1n + \cdots + \frac1n$$ does, can't be handled this way. | {
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To summarize: Many of the limit laws feel like they're just saying "take the limit of the parts of the expression". This isn't true; in fact, they're saying "here is one precise way in which you can find a limit by using the limits of the parts". If you want to do something to a limit that isn't one of the standard limit laws, you're doing something special, which means you'll need to go back to the definition of the limit (or something similar) in order to make sure that what you're doing works.
• Excellent answer. I don't know if this is worth noting, but we could peel off one term (or finitely many) from the expression in question - we just are always left with a sum that doesn't have a fixed size. e.g. $\displaystyle{\lim_{n\to\infty}}\underbrace{\frac1n+\frac1n+\cdots}_{n\text{ terms}}=\displaystyle{\lim_{n\to\infty}}\frac1n+\displaystyle{\lim_{n\to\infty}}\underbrace{\frac1n+\frac1n+\cdots}_{n-1\text{ terms}}$ – Mark S. Aug 5 at 13:05 | {
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# Checking if a symbolic symmetrical Matrix is negative definite
I have the following problem finding the value ranges for the parameters of a symbolic symmetrical matrix in order to make it negative definite:
The matrix I'm talking about looks as follows
A := {{-1, -b, a, 0}, {-b, -1, 0, 0}, {a, 0, -a, -b a}, {0, 0, -b a, -a}}
and as you can see in matrix form, it is a symmetrical matrix
\begin{array}{cccc} -1 & -b & a & 0 \\ -b & -1 & 0 & 0 \\ a & 0 & -a & -a b \\ 0 & 0 & -a b & -a \\ \end{array}
Now I'm trying to find the value ranges of a and b in order to make the matrix negative definite. It is important that b depends on a and not the other way round, since the matrix is part of an economic model, which doesn't make any sense otherwise.
First I used the approach to find the value ranges, which make all Eigenvalues negative and thus lead to a negative definite matrix
Reduce[Eigenvalues[A] < 0, {a, b}]
which yields
0 < a < 1 && -Sqrt[1 - Sqrt[a]] < b < Sqrt[1 - Sqrt[a]]
Everything fine so far. But then I tried a different approach. If the k-th order leading principal minor of the matrix has sign (-1)^k, then the matrix should be negative definite, so I'm expecting the same result:
A1 := {{-1}}
A2 := {{-1, -b}, {-b, -1}}
A3 := {{-1, -b, a}, {-b, -1, 0}, {a, 0, -a}}
Reduce[{Det[A1] < 0, Det[A2] > 0, Det[A3] < 0, Det[A] > 0}, {a, b}]
which yields
0 < a < 1 && Root[1 - a - 2 #1^2 + #1^4 &, 2] < b < Root[1 - a - 2 #1^2 + #1^4 &, 3]
ToRadicals[
0 < a < 1 && Root[1 - a - 2 #1^2 + #1^4 &, 2] < b < Root[1 - a - 2 #1^2 + #1^4 &, 3]]
0 < a < 1 && -Sqrt[1 - Sqrt[a]] < b < Sqrt[1 + Sqrt[a]]
As you can see, the result is different than in the first approach (to be more specific the upper bound of b is different), which makes no sense, since both approaches should yield the same result.
Does anyone know what I am doing wrong or which of the results is correct?
Thanks a lot,
Phil | {
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Does anyone know what I am doing wrong or which of the results is correct?
Thanks a lot,
Phil
• The latter answer is incorrect: A /. {a -> 1/2, b -> Sqrt[1 + Sqrt[1/2]]} // Eigenvalues // N has a positive eigenvalue. Maybe you set up the deterimnants incorrectly since it looks like a sign error. – bill s Aug 11 '17 at 18:51
• See @Szabolcs' answer to a question about reordering when using ToRadicals on Root objects. – Carl Woll Aug 11 '17 at 19:27
• Thanks @CarlWoll, I think the ordering of the Root objects is on the right track. But I still haven't found a solution for my specific case – PTSammy Aug 13 '17 at 12:56
Both of your approaches yield the same answer. it is the application of ToRadicals that causes the answer to be different. First, compare the two limits before the application of ToRadicals:
upperLimit1 = Sqrt[1 - Sqrt[a]];
upperLimit2 = Root[1 - a - 2 #1^2 + #1^4 &, 3];
Plot[upperLimit1, {a, 0, 1}]
Plot[upperLimit2, {a, 0, 1}]
upperLimit1 and upperLimit2 are the same over the region 0 < a < 1. Converting the Root object into radicals is problematic because the Root ordering depends on the parameter a. One suggestion would be to not use ToRadicals and just work with the Root objects. If you really want radicals, a naive application of ToRadicals:
ToRadicals[upperLimit2]
Sqrt[1 + Sqrt[a]]
is only correct for some values of the parameter a. However, in your case, you know something about the parameter a, so you should make use of that by giving ToRadicals an assumption:
ToRadicals[upperLimit2, Assumptions -> 0 < a < 1]
Sqrt[1 - Sqrt[a]]
Note that using:
ToRadicals[0 < a < 1 && upperLimit2]
0 < a < 1 && Sqrt[1 + Sqrt[a]]
does not cause ToRadicals to use 0 < a < 1 as an assumption. The assumption needs to be given explicitly as an option to ToRadicals.
In this answer I will briefly go through the method of using the leading principal minors to derive negative definiteness. Hope this helps you verify your results | {
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