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college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.2.11
|
Suppose men's heights (in centimeters) follow the distribution $N\left(174,20^{2}\right)$, while those of women follow the distribution $N\left(160,15^{2}\right)$. Compute the mean total height of a man-woman married couple.
|
334
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
334
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.2.11
|
Suppose $S=\{1,2,3\}$, and $P(\{1\})=P(\{2\})+1 / 6$, and $P(\{3\})=2 P(\{2\})$. Compute $P(\{1\}), P(\{2\})$, and $P(\{3\})$.
|
$P(\{2\})=5 / 24, P(\{1\})=3 / 8, P(\{3\})=5 / 12$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P(\{2\})=5 / 24, P(\{1\})=3 / 8, P(\{3\})=5 / 12
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.4.7
|
Let $Y \sim$ Poisson $(\lambda)$. Compute $E\left(Y^{3}\right)$, the third moment of $Y$.
|
$m_{Y}^{\prime \prime \prime}(s)=e^{\lambda\left(e^{s}-1\right)} e^{s} \lambda\left(1+3 e^{s} \lambda+e^{2 s} \lambda^{2}\right), E\left(Y^{3}\right)=m_{Y}^{\prime \prime \prime}(0)=\lambda\left(1+3 \lambda+\lambda^{2}\right)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
m_{Y}^{\prime \prime \prime}(s)=e^{\lambda\left(e^{s}-1\right)} e^{s} \lambda\left(1+3 e^{s} \lambda+e^{2 s} \lambda^{2}\right), E\left(Y^{3}\right)=m_{Y}^{\prime \prime \prime}(0)=\lambda\left(1+3 \lambda+\lambda^{2}\right)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.2.11
|
Suppose that a chi-squared test is carried out, based on a random sample of $n$ from a population, to assess whether or not two categorical variables $X$ and $Y$ are independent. Suppose the P-value equals 0.001 and the investigator concludes that there is evidence against independence. Discuss how you would check to see if the deviation from independence was of practical significance.
|
We look at the differences $\left|f_{i j}-f i \cdot f_{\cdot j} / n\right|$ to see how big these are.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
We look at the differences $\left|f_{i j}-f i \cdot f_{\cdot j} / n\right|$ to see how big these are.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.6.5
|
Let $X \sim \operatorname{Exponential}(\lambda)$. Let $Y=X^{3}$. Compute the density $f_{Y}$ of $Y$.
|
$f_{Y}(y)$ equals $(\lambda / 3) y^{-2 / 3} e^{-\lambda y^{1 / 3}}$ for $y>0$ and otherwise equals 0 .
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
f_{Y}(y)$ equals $(\lambda / 3) y^{-2 / 3} e^{-\lambda y^{1 / 3}}$ for $y>0$ and otherwise equals 0 .
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.1.5
|
Suppose that a symmetrical die is tossed $n=20$ independent times. Work out the exact sampling distribution of the maximum of this sample.
|
For $1 \leq j \leq 6, P(\max =j)=(j / 6)^{20}-((j-1) / 6)^{20}$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
For $1 \leq j \leq 6, P(\max =j)=(j / 6)^{20}-((j-1) / 6)^{20}$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.5.11
|
Suppose that a statistical model is given by the family of $N\left(\mu, \sigma^{2}\right)$ distributions where $\theta=\left(\mu, \sigma^{2}\right) \in R^{1} \times R^{+}$is unknown. If our interest is in making inferences about the distribution function evaluated at 3 , then determine $\psi\left(\mu, \sigma^{2}\right)$.
|
$\psi\left(\mu, \sigma^{2}\right)=\Phi((3-\mu) / \sigma)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\psi\left(\mu, \sigma^{2}\right)=\Phi((3-\mu) / \sigma)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.11.4.3
|
Suppose we define a process $\left\{X_{n}\right\}$ as follows. Given $X_{n}$, with probability $p$ we let $X_{n+1}=2 X_{n}$, while with probability $1-p$ we let $X_{n+1}=X_{n} / 2$. What value of $p$ will make $\left\{X_{n}\right\}$ be a martingale?
|
$p=1 / 3$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
p=1 / 3
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.4.7
|
Suppose we keep dealing cards from an ordinary 52-card deck until the first jack appears. What is the probability that at least 10 cards go by before the first jack?
|
$\left(\begin{array}{c}48 \\ 10\end{array}\right) /\left(\begin{array}{c}52 \\ 10\end{array}\right)=246 / 595=0.4134$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\left(\begin{array}{c}48 \\ 10\end{array}\right) /\left(\begin{array}{c}52 \\ 10\end{array}\right)=246 / 595=0.4134
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.11.5.9
|
Let $X_{t}=10-1.5 t+4 B_{t}$. Compute $E\left(X_{3} X_{5}\right)$.
|
$E\left(X_{3} X_{5}\right)=61.75$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E\left(X_{3} X_{5}\right)=61.75
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.4.3
|
Suppose we flip 100 fair independent coins. What is the probability that at least three of them are heads? (Hint: You may wish to use (1.3.1).)
|
$1-5051 / 2^{100}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
1-5051 / 2^{100}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.4.5
|
Verify that the third moment of an $N\left(\mu, \sigma^{2}\right)$ distribution is given by $\mu_{3}=$ $\mu^{3}+3 \mu \sigma^{2}$. Because the normal distribution is specified by its first two moments, any characteristic of the normal distribution can be estimated by simply plugging in the MLE estimates of $\mu$ and $\sigma^{2}$. Compare the method of moments estimator of $\mu_{3}$ with this plug-in MLE estimator, i.e., determine whether they are the same or not.
|
From the mgf, $m_{X}^{\prime \prime \prime}(0)=3 \sigma^{2} \mu+\mu^{3}$. The plug-in estimator is $\hat{\mu}_{3}=3\left(m_{2}-m_{1}^{2}\right) \times$ $m_{1}+m_{1}^{3}$, while the method of moments estimator of $\mu_{3}$ is $m_{3}=\frac{1}{n} \sum x_{i}^{3}$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
From the mgf, $m_{X}^{\prime \prime \prime}(0)=3 \sigma^{2} \mu+\mu^{3}$. The plug-in estimator is $\hat{\mu}_{3}=3\left(m_{2}-m_{1}^{2}\right) \times$ $m_{1}+m_{1}^{3}$, while the method of moments estimator of $\mu_{3}$ is $m_{3}=\frac{1}{n} \sum x_{i}^{3}$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.1.5
|
The following data were generated from an $N(\mu, 1)$ distribution by a student. Unfortunately, the student forgot which value of $\mu$ was used, so we are uncertain about the correct probability distribution to use to describe the variation in the data.
\begin{tabular}{rrrrrrrr|}
\hline 0.2 & -0.7 & 0.0 & -1.9 & 0.7 & -0.3 & 0.3 & 0.4 \\
0.3 & -0.8 & 1.5 & 0.1 & 0.3 & -0.7 & -1.8 & 0.2 \\
\hline
\end{tabular}
Can you suggest a plausible value for $\mu$ ? Explain your reasoning.
|
$\bar{x}=-0.1375$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\bar{x}=-0.1375
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.2.11
|
Suppose you are measuring the volume of a cubic box in centimeters by taking repeated independent measurements of one of the sides. Suppose it is reasonable to assume that a single measurement follows an $N\left(\mu, \sigma_{0}^{2}\right)$ distribution, where $\mu$ is unknown and $\sigma_{0}^{2}$ is known. Based on a sample of measurements, you obtain the MLE of $\mu$ as 3.2 $\mathrm{cm}$. What is your estimate of the volume of the box? How do you justify this in terms of the likelihood function?
|
$\hat{\mu}^{3}=32.768 \mathrm{~cm}^{3}$ is the MLE
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\hat{\mu}^{3}=32.768 \mathrm{~cm}^{3}$ is the MLE
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.3.13
|
Suppose that a simple linear model is fit to data. An analysis of the residuals indicates that there is no reason to doubt that the model is correct; the ANOVA test indicates that there is substantial evidence against the null hypothesis of no relationship between the response and predictor. The value of $R^{2}$ is found to be 0.05 . What is the interpretation of this number and what are the practical consequences?
|
$R^{2}=0.05$ indicates that the linear model explains only $5 \%$ of the variation in the response, so the model will not have much predictive power.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
R^{2}=0.05$ indicates that the linear model explains only $5 \%$ of the variation in the response, so the model will not have much predictive power.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.3.1
|
Suppose there are three coins - one is known to be fair, one has probability $1 / 3$ of yielding a head on a single toss, and one has probability $2 / 3$ for head on a single toss. A coin is selected (not randomly) and then tossed five times. The goal is to make an inference about which of the coins is being tossed, based on the sample. Fully describe a statistical model for a single response and for the sample.
|
The statistical model for a single response consists of three probability functions \{Bernoulli(1/2), Bernoulli(1/3), Bernoulli(2/3)\}.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The statistical model for a single response consists of three probability functions \{Bernoulli(1/2), Bernoulli(1/3), Bernoulli(2/3)\}.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.1.7
|
Let $X \sim \operatorname{Binomial}(80,1 / 4)$, and let $Y \sim \operatorname{Poisson}(3 / 2)$. Assume $X$ and $Y$ are independent. Compute $E(X Y)$.
|
$E(X Y)=30$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E(X Y)=30
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.3.3
|
Marks on an exam in a statistics course are assumed to be normally distributed with unknown mean but with variance equal to 5 . A sample of four students is selected, and their marks are $52,63,64,84$. Assess the hypothesis $H_{0}: \mu=60$ by computing the relevant $\mathrm{P}$-value and compute a 0.95 -confidence interval for the unknown $\mu$.
|
P-value $=0.000$ and 0.95 -confidence interval is $(63.56,67.94)$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P-value $=0.000$ and 0.95 -confidence interval is $(63.56,67.94)$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.4.9
|
Suppose two measurements, $Y_{1}$ and $Y_{2}$, corresponding to different treatments, are taken on the same individual who has been randomly sampled from a population $\Pi$. Suppose that $Y_{1}$ and $Y_{2}$ have the same variance and are negatively correlated. Our goal is to compare the treatment means. Explain why it would have been better to have randomly sampled two individuals from $\Pi$ and applied the treatments to these individuals separately. (Hint: Consider $\operatorname{Var}\left(Y_{1}-Y_{2}\right)$ in these two sampling situations.)
|
When $Y_{1}$ and $Y_{2}$ are measured on the same individual, we have that $\operatorname{Var}\left(Y_{1}-\right.$ $\left.Y_{2}\right)=2\left(\operatorname{Var}\left(Y_{1}\right)-\operatorname{Cov}\left(Y_{1}, Y_{2}\right)\right)>2 \operatorname{Var}\left(Y_{1}\right)$ since $\operatorname{Cov}\left(Y_{1}, Y_{2}\right)<0$. If we had measured $Y_{1}$ and $Y_{2}$ on independently randomly selected individuals, then we would have that $\operatorname{Var}\left(Y_{1}-Y_{2}\right)=2 \operatorname{Var}\left(Y_{1}\right)$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
When $Y_{1}$ and $Y_{2}$ are measured on the same individual, we have that $\operatorname{Var}\left(Y_{1}-\right.$ $\left.Y_{2}\right)=2\left(\operatorname{Var}\left(Y_{1}\right)-\operatorname{Cov}\left(Y_{1}, Y_{2}\right)\right)>2 \operatorname{Var}\left(Y_{1}\right)$ since $\operatorname{Cov}\left(Y_{1}, Y_{2}\right)<0$. If we had measured $Y_{1}$ and $Y_{2}$ on independently randomly selected individuals, then we would have that $\operatorname{Var}\left(Y_{1}-Y_{2}\right)=2 \operatorname{Var}\left(Y_{1}\right)$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.2.3
|
Suppose $S=\{1,2,3\}$, with $P(\{1\})=1 / 2$ and $P(\{1,2\})=2 / 3$. What must $P(\{2\})$ be?
|
$P(\{2\})=1 / 6$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P(\{2\})=1 / 6
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.7.4.9
|
Suppose a student wants to put a prior on the mean grade out of 100 that their class will obtain on the next statistics exam. The student feels that a normal prior centered at 66 is appropriate and that the interval $(40,92)$ should contain $99 \%$ of the marks. Fully identify the prior.
|
The prior distribution is $\theta \sim N\left(66, \sigma^{2}\right)$ with $\sigma^{2}=101.86$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The prior distribution is $\theta \sim N\left(66, \sigma^{2}\right)$ with $\sigma^{2}=101.86$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.7.9
|
Suppose the cdf of $W$ is given by $F_{W}(w)=0$ for $w<10, F_{W}(w)=w-10$ for $10 \leq w \leq 11$, and by $F_{W}(w)=1$ for $w>11$. Compute $E(W)$. (Hint: Remember that $F_{W}(w)=P(W \leq w)=1-P(W>w)$.)
|
$E(W)=21 / 2$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E(W)=21 / 2
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.8.1.3
|
Suppose that $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from an $N\left(\mu, \sigma_{0}^{2}\right)$ distribution, where $\mu \in$ $R^{1}$ is unknown and $\sigma_{0}^{2}$ is known. Determine a UMVU estimator of the second moment $\mu^{2}+\sigma_{0}^{2}$.
|
$\bar{x}^{2}+(1-1 / n) \sigma_{0}^{2}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\bar{x}^{2}+(1-1 / n) \sigma_{0}^{2}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.6.5
|
Let $W \sim \operatorname{Binomial}(100,1 / 2)$, as in the number of heads when flipping 100 fair coins. Use Chebychev's inequality to get an upper bound on $P(|W-50| \geq 10)$.
|
$1 / 4$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
1 / 4
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.3.17
|
A P-value was computed to assess the hypothesis $H_{0}: \psi(\theta)=0$ and the value 0.22 was obtained. The investigator says this is strong evidence that the hypothesis is correct. How do you respond?
|
The P-value 0.22 does not imply the null hypothesis is correct. It may be that we have just not taken a large enough sample size to detect a difference.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The P-value 0.22 does not imply the null hypothesis is correct. It may be that we have just not taken a large enough sample size to detect a difference.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.11.6.3
|
Let $\{N(t)\}_{t \geq 0}$ be a Poisson process with intensity $a=1 / 3$. Compute $P\left(N_{2}=\right.$ 6) and $P\left(N_{3}=5\right)$.
|
$P\left(N_{2}=6\right)=e^{-2 / 3}(2 / 3)^{6} / 6$ !, $P\left(N_{3}=5\right)=e^{-3 / 3}(3 / 3)^{5} / 5$ !
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P\left(N_{2}=6\right)=e^{-2 / 3}(2 / 3)^{6} / 6$ !, $P\left(N_{3}=5\right)=e^{-3 / 3}(3 / 3)^{5} / 5$ !
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.2.9
|
Suppose that $X \sim \operatorname{Geometric}(1 / 3)$. What value would you record as a prediction of a future value of $X$ ?
|
The mode is $x=0$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The mode is $x=0$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.2.15
|
If two functions of $\theta$ are equivalent versions of the likelihood when one is a positive multiple of the other, then when are two log-likelihood functions equivalent?
|
Equivalent log-likelihood functions differ by an additive constant.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
Equivalent log-likelihood functions differ by an additive constant.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.5.13
|
Suppose that a statistical model is given by the family of $\operatorname{Bernoulli}(\theta)$ distributions where $\theta \in \Omega=[0,1]$. If our interest is in making inferences about the probability that in two independent observations from this model we obtain a 0 and a 1 , then determine $\psi(\theta)$.
|
$\psi(\theta)=2 \theta(1-\theta)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\psi(\theta)=2 \theta(1-\theta)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.5.9
|
Suppose a certain experiment has probability $\theta$ of success, where $0<\theta<1$ but $\theta$ is unknown. Suppose we repeat the experiment 1000 times, of which 400 are successes and 600 are failures. Compute an interval of values that are virtually certain to contain $\theta$.
|
$(0.354,0.447)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
(0.354,0.447)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.2.7
|
Suppose a university is composed of $55 \%$ female students and $45 \%$ male students. A student is selected to complete a questionnaire. There are 25 questions on the questionnaire administered to a male student and 30 questions on the questionnaire administered to a female student. If $X$ denotes the number of questions answered by a randomly selected student, then compute $P(X=x)$ for every real number $x$.
|
$P(X=25)=0.45, P(X=30)=0.55$, and $P(X=x)=0$ otherwise
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P(X=25)=0.45, P(X=30)=0.55$, and $P(X=x)=0$ otherwise
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.1.5
|
Let $A$ and $B$ be events, and let $X=I_{A} \cdot I_{B}$. Is $X$ an indicator function? If yes, then of what event?
|
Yes, for $A \cap B$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
Yes, for $A \cap B$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.3.11
|
Suppose a possibly biased die is rolled 30 times and that the face containing two pips comes up 10 times. Do we have evidence to conclude that the die is biased?
|
P-value $=0.014$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P-value $=0.014
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.8.1.5
|
Suppose that $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from an $N\left(\mu, \sigma_{0}^{2}\right)$ distribution, where $\mu \in$ $R^{1}$ is unknown and $\sigma_{0}^{2}$ is known. Is $2 \bar{x}+3$ a UMVU estimator of anything? If so, what is it UMVU for? Justify your answer.
|
UMVU for $5+2 \mu$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
UMVU for $5+2 \mu
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.3.7
|
Suppose your team has a $40 \%$ chance of winning or tying today's game and has a $30 \%$ chance of winning today's game. What is the probability that today's game will be a tie?
|
$10 \%$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
10 \%
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.6.13
|
Suppose a species of beetle has length 35 millimeters on average. Find an upper bound on the probability that a randomly chosen beetle of this species will be over 80 millimeters long.
|
$7 / 16$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
7 / 16
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.5.9
|
Suppose that a statistical model is given by the family of $N\left(\mu, \sigma_{0}^{2}\right)$ distributions where $\theta=\mu \in R^{1}$ is unknown, while $\sigma_{0}^{2}$ is known. If our interest is in making inferences about the distribution function evaluated at 3, then determine $\psi(\mu)$.
|
$\psi(\mu)=\Phi\left((3-\mu) / \sigma_{0}\right)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\psi(\mu)=\Phi\left((3-\mu) / \sigma_{0}\right)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.1.5
|
Suppose that $X$ is a random variable and $Y=X^{2}$. Determine whether or not $X$ and $Y$ are related. What happens when $X$ has a degenerate distribution?
|
The conditional distributions $P(Y=y \mid X=x)$ will change with $x$ whenever $X$ is not degenerate.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The conditional distributions $P(Y=y \mid X=x)$ will change with $x$ whenever $X$ is not degenerate.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.2.5
|
Suppose that $X \sim N(10,2)$. What value would you record as a prediction of a future value of $X$ ? How would you justify your choice?
|
$x=10$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
x=10
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.3.1
|
Suppose measurements (in centimeters) are taken using an instrument. There is error in the measuring process and a measurement is assumed to be distributed $N\left(\mu, \sigma_{0}^{2}\right)$, where $\mu$ is the exact measurement and $\sigma_{0}^{2}=0.5$. If the $(n=10)$ measurements 4.7, 5.5, 4.4, 3.3, 4.6, 5.3, 5.2, 4.8, 5.7, 5.3 were obtained, assess the hypothesis $H_{0}: \mu=5$ by computing the relevant P-value. Also compute a 0.95 -confidence interval for the unknown $\mu$.
|
$\mathrm{P}$-value $=0.592$ and 0.95 -confidence interval is $(4.442,5.318)$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\mathrm{P}$-value $=0.592$ and 0.95 -confidence interval is $(4.442,5.318)$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.1.19
|
Suppose a variable $X$ takes the values 1 and 2 on a population and the conditional distributions of $Y$ given $X$ are $N(0,5)$ when $X=1$, and $N(0,7)$ when $X=2$. Determine whether $X$ and $Y$ are related and if so, describe their relationship.
|
$X$ and $Y$ are related. We see that only the variance of the conditional distribution changes as we change $X$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
X$ and $Y$ are related. We see that only the variance of the conditional distribution changes as we change $X$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.3.3
|
Suppose that $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from the Exponential $(\theta)$, where $\theta>0$ is unknown. What is the least-squares estimate of the mean of this distribution?
|
$\bar{x}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\bar{x}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.2.1
|
Consider flipping two independent fair coins. Let $X$ be the number of heads that appear. Compute $P(X=x)$ for all real numbers $x$.
|
$P(X=0)=P(X=2)=1 / 4, P(X=1)=1 / 2, P(X=x)=0$ for $x \neq 0,1,2$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P(X=0)=P(X=2)=1 / 4, P(X=1)=1 / 2, P(X=x)=0$ for $x \neq 0,1,2
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.3.7
|
Let $X \sim \operatorname{Binomial}(12, \theta)$. For what value of $\theta$ is $P(X=11)$ maximized?
|
$\theta=11 / 12$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\theta=11 / 12
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.4.5
|
Is the function defined by $f(x)=x / 3$ for $-1<x<2$ and 0 otherwise, a density? Why or why not?
|
No
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
No
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.7.1.7
|
Suppose we have a sample
\begin{tabular}{|llllllllll|}
\hline 6.56 & 6.39 & 3.30 & 3.03 & 5.31 & 5.62 & 5.10 & 2.45 & 8.24 & 3.71 \\
4.14 & 2.80 & 7.43 & 6.82 & 4.75 & 4.09 & 7.95 & 5.84 & 8.44 & 9.36 \\
\hline
\end{tabular}
from an $N\left(\mu, \sigma^{2}\right)$ distribution and we determine that a prior specified by $\mu \mid \sigma^{2} \sim$ $N\left(3,4 \sigma^{2}\right), \sigma^{-2} \sim \operatorname{Gamma}(1,1)$ is appropriate. Determine the posterior distribution of $\left(\mu, 1 / \sigma^{2}\right)$.
|
$\mu\left|\sigma^{2}, x_{1}, \ldots, x_{n} \sim N\left(5.5353, \frac{4}{81} \sigma^{2}\right), 1 / \sigma^{2}\right| x_{1}, \ldots, x_{n} \sim \operatorname{Gamma}(11,41.737)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\mu\left|\sigma^{2}, x_{1}, \ldots, x_{n} \sim N\left(5.5353, \frac{4}{81} \sigma^{2}\right), 1 / \sigma^{2}\right| x_{1}, \ldots, x_{n} \sim \operatorname{Gamma}(11,41.737)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.3.3
|
Suppose that an employee arrives late $10 \%$ of the time, leaves early $20 \%$ of the time, and both arrives late and leaves early $5 \%$ of the time. What is the probability that on a given day that employee will either arrive late or leave early (or both)?
|
$P$ (late or early or both) $=25 \%$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P$ (late or early or both) $=25 \%
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.2.7
|
Write out in full how you would generate a value from a Dirichlet $(1,1,1,1)$ distribution.
|
We should first generate a value for $X_{1} \sim \operatorname{Dirichlet}(1,3)$. Then generate $U_{2}$ from the $\operatorname{Beta}(1,2)$ distribution and set $X_{2}=\left(1-X_{1}\right) U_{2}$. Next generate $U_{3}$ from the $\operatorname{Beta}(1,1)$ distribution and set $X_{3}=\left(1-X_{1}-X_{2}\right) U_{3}$. Finally, set $X_{4}=1-X_{1}-$ $X_{2}-X_{3}$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
We should first generate a value for $X_{1} \sim \operatorname{Dirichlet}(1,3)$. Then generate $U_{2}$ from the $\operatorname{Beta}(1,2)$ distribution and set $X_{2}=\left(1-X_{1}\right) U_{2}$. Next generate $U_{3}$ from the $\operatorname{Beta}(1,1)$ distribution and set $X_{3}=\left(1-X_{1}-X_{2}\right) U_{3}$. Finally, set $X_{4}=1-X_{1}-$ $X_{2}-X_{3}$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.2.3
|
If $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from a $\operatorname{Bernoulli}(\theta)$ distribution, where $\theta \in[0,1]$ is unknown, then determine the MLE of $\theta^{2}$.
|
$\psi(\theta)=\theta^{2}$ is $1-1$, and so $\psi\left(\hat{\theta}\left(x_{1}, \ldots, x_{n}\right)\right)=\bar{x}^{2}$ is the MLE.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\psi(\theta)=\theta^{2}$ is $1-1$, and so $\psi\left(\hat{\theta}\left(x_{1}, \ldots, x_{n}\right)\right)=\bar{x}^{2}$ is the MLE.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.4.12
|
Determine if the system is consistent. If so, is the solution unique?
$$
\begin{gathered}
x+2 y+z-w=2 \\
x-y+z+w=1 \\
2 x+y-z=1 \\
4 x+2 y+z=5
\end{gathered}
$$
|
There is no solution.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
There is no solution.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.5.19
|
Suppose that $A, B, C, D$ are $n \times n$-matrices, and that all relevant matrices are invertible. Further, suppose that $(A+B)^{-1}=C B^{-1}$. Solve this equation for $A$ (in terms of $B$ and $C$ ), $B$ (in terms of $A$ and $C$ ), and $C$ (in terms of $A$ and $B$ ).
|
To solve for $A$, we invert both sides of the equation $(A+B)^{-1}=C B^{-1}$ and use matrix algebra to get $A+B=\left(C B^{-1}\right)^{-1}=\left(B^{-1}\right)^{-1} C^{-1}=B C^{-1}$. Therefore, $A=B C^{-1}-B$.
To solve for $B$, we note that $A=B C^{-1}-B=B\left(C^{-1}-I\right)$. Multiplying both sides of the equation on the right by the inverse of $C^{-1}-I$, we get $B=A\left(C^{-1}-I\right)^{-1}$.
To solve for $C$, we take the original equation $(A+B)^{-1}=C B^{-1}$ and right-multiply both sides of the equation $B$. This yields $C=(A+B)^{-1} B$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
To solve for $A$, we invert both sides of the equation $(A+B)^{-1}=C B^{-1}$ and use matrix algebra to get $A+B=\left(C B^{-1}\right)^{-1}=\left(B^{-1}\right)^{-1} C^{-1}=B C^{-1}$. Therefore, $A=B C^{-1}-B$.
To solve for $B$, we note that $A=B C^{-1}-B=B\left(C^{-1}-I\right)$. Multiplying both sides of the equation on the right by the inverse of $C^{-1}-I$, we get $B=A\left(C^{-1}-I\right)^{-1}$.
To solve for $C$, we take the original equation $(A+B)^{-1}=C B^{-1}$ and right-multiply both sides of the equation $B$. This yields $C=(A+B)^{-1} B$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.5.2.5
|
Are the following vectors linearly independent? If not, write one of them as a linear combination of the others.
$$
\mathbf{u}=\left[\begin{array}{l}
1 \\
3 \\
1
\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}
1 \\
4 \\
2
\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}
1 \\
1 \\
-1
\end{array}\right]
$$
|
The vectors are linearly dependent. We have $\mathbf{w}=3 \mathbf{u}-2 \mathbf{v}$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
The vectors are linearly dependent. We have $\mathbf{w}=3 \mathbf{u}-2 \mathbf{v}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.7.18
|
Simplify $\|\mathbf{u} \times \mathbf{v}\|^{2}+(\mathbf{u} \cdot \mathbf{v})^{2}-\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}$.
|
We have
$$
\begin{aligned}
\|\mathbf{u} \times \mathbf{v}\|^{2} & =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2} \sin ^{2} \theta \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}\left(1-\cos ^{2} \theta\right) \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2} \cos ^{2} \theta \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}
\end{aligned}
$$
which implies the expression equals 0 .
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
We have
$$
\begin{aligned}
\|\mathbf{u} \times \mathbf{v}\|^{2} & =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2} \sin ^{2} \theta \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}\left(1-\cos ^{2} \theta\right) \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2} \cos ^{2} \theta \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}
\end{aligned}
$$
which implies the expression equals 0 .
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.7.4
|
Find the area of the parallelogram determined by the vectors $\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right],\left[\begin{array}{r}4 \\ -2 \\ 1\end{array}\right]$.
|
$\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] \times\left[\begin{array}{r}4 \\ -2 \\ 1\end{array}\right]=\left[\begin{array}{r}6 \\ 11 \\ -2\end{array}\right]$. The area is of the parallelogram is $\sqrt{36+121+4}=\sqrt{161}$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] \times\left[\begin{array}{r}4 \\ -2 \\ 1\end{array}\right]=\left[\begin{array}{r}6 \\ 11 \\ -2\end{array}\right]$. The area is of the parallelogram is $\sqrt{36+121+4}=\sqrt{161}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.4.22
|
Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.
|
No. Consider $x+y+z=2$ and $x+y+z=1$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
No. Consider $x+y+z=2$ and $x+y+z=1$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.7.3
|
Which of the following matrices are symmetric, antisymmetric, both, or neither?
$$
A=\left[\begin{array}{rr}
0 & 1 \\
-1 & 0
\end{array}\right], \quad B=\left[\begin{array}{ll}
2 & 1 \\
1 & 3
\end{array}\right], \quad C=\left[\begin{array}{rr}
1 & 2 \\
-2 & 0
\end{array}\right], \quad D=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right] .
$$
|
$A$ is antisymmetric, $B$ is symmetric, $C$ is neither, and $D$ is both.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
A$ is antisymmetric, $B$ is symmetric, $C$ is neither, and $D$ is both.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.8.4.2
|
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
-13 & -28 & 28 \\
4 & 9 & -8 \\
-4 & -8 & 9
\end{array}\right]
$$
One eigenvalue is 3. Diagonalize if possible.
|
The eigenvectors and eigenvalues are:
$$
\left\{\left[\begin{array}{l}
2 \\
0 \\
1
\end{array}\right],\left[\begin{array}{c}
-2 \\
1 \\
0
\end{array}\right]\right\} \text { for eigenvalue } 1, \quad\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \text { for eigenvalue } 3 .
$$
The matrix $P$ needed to diagonalize the above matrix is
$$
\left[\begin{array}{rrr}
2 & -2 & 7 \\
0 & 1 & -2 \\
1 & 0 & 2
\end{array}\right]
$$
and the diagonal matrix $D$ is
$$
\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{array}\right]
$$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
The eigenvectors and eigenvalues are:
$$
\left\{\left[\begin{array}{l}
2 \\
0 \\
1
\end{array}\right],\left[\begin{array}{c}
-2 \\
1 \\
0
\end{array}\right]\right\} \text { for eigenvalue } 1, \quad\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \text { for eigenvalue } 3 .
$$
The matrix $P$ needed to diagonalize the above matrix is
$$
\left[\begin{array}{rrr}
2 & -2 & 7 \\
0 & 1 & -2 \\
1 & 0 & 2
\end{array}\right]
$$
and the diagonal matrix $D$ is
$$
\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.5.2.10
|
Here are some vectors.
$$
\mathbf{u}_{1}=\left[\begin{array}{r}
1 \\
2 \\
-2
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{r}
2 \\
2 \\
-4
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{r}
2 \\
7 \\
-4
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{r}
5 \\
7 \\
-10
\end{array}\right], \quad \mathbf{u}_{5}=\left[\begin{array}{r}
12 \\
17 \\
-24
\end{array}\right] .
$$
Describe the span of these vectors as the span of as few vectors as possible.
|
$$
\left[\begin{array}{rrrrr}
1 & 2 & 2 & 5 & 12 \\
1 & 2 & 7 & 7 & 17 \\
-2 & -4 & -4 & -10 & -24
\end{array}\right] \simeq\left[\begin{array}{llllr}
1 & 1 & 2 & 5 & 12 \\
0 & 0 & 5 & 2 & 5 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
Linearly independent subset: $\left\{\mathbf{u}_{1}, \mathbf{u}_{3}\right\}$. Since the rank is 2 , this is the smallest possible.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\left[\begin{array}{rrrrr}
1 & 2 & 2 & 5 & 12 \\
1 & 2 & 7 & 7 & 17 \\
-2 & -4 & -4 & -10 & -24
\end{array}\right] \simeq\left[\begin{array}{llllr}
1 & 1 & 2 & 5 & 12 \\
0 & 0 & 5 & 2 & 5 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
Linearly independent subset: $\left\{\mathbf{u}_{1}, \mathbf{u}_{3}\right\}$. Since the rank is 2 , this is the smallest possible.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.1.3
|
What are the dimensions of the following matrices?
$$
A=\left[\begin{array}{rrr}
1 & -2 & 0 \\
4 & 3 & 2
\end{array}\right], \quad B=\left[\begin{array}{lll}
3 & 4 & 1 \\
1 & 3 & 1 \\
6 & 2 & 2
\end{array}\right], \quad C=\left[\begin{array}{ll}
1 & 0 \\
4 & 0 \\
2 & 0 \\
0 & 0
\end{array}\right]
$$
|
$2 \times 3,3 \times 3,4 \times 2$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
2 \times 3,3 \times 3,4 \times 2$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.3.1.6
|
Consider the following vector equation for a line in $\mathbb{R}^{3}$ :
$$
\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
1 \\
2 \\
0
\end{array}\right]+t\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]
$$
Find a new vector equation for the same line by doing the change of parameter $t=2-s$.
|
We have
$$
\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
1 \\
2 \\
0
\end{array}\right]+(2-s)\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]=\left[\begin{array}{l}
3 \\
2 \\
2
\end{array}\right]+s\left[\begin{array}{r}
-1 \\
0 \\
-1
\end{array}\right] .
$$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
We have
$$
\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
1 \\
2 \\
0
\end{array}\right]+(2-s)\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]=\left[\begin{array}{l}
3 \\
2 \\
2
\end{array}\right]+s\left[\begin{array}{r}
-1 \\
0 \\
-1
\end{array}\right] .
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.5.3
|
Use Gauss-Jordan elimination to solve the system of equations $-8 x+2 y+5 z=18,-8 x+$ $3 y+5 z=13$, and $-4 x+y+5 z=19$.
|
Solution is: $[x=-1, y=-5, z=4]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Solution is: $[x=-1, y=-5, z=4]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.10.7
|
Which of the following vectors, if any, are orthogonal?
$\mathbf{u}_{1}=\left[\begin{array}{c}1 \\ i \\ 0\end{array}\right]$,
$\mathbf{u}_{2}=\left[\begin{array}{c}1 \\ -i \\ 1\end{array}\right]$,
$\mathbf{u}_{3}=\left[\begin{array}{c}0 \\ 1 \\ -i\end{array}\right]$,
$\mathbf{u}_{4}=\left[\begin{array}{c}0 \\ 0 \\ 1\end{array}\right]$.
|
$\mathbf{u}_{1} \perp \mathbf{u}_{2}, \mathbf{u}_{1} \perp \mathbf{u}_{4}$, and $\mathbf{u}_{2} \perp \mathbf{u}_{3}$.
|
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college_math.linear_algebra
|
\mathbf{u}_{1} \perp \mathbf{u}_{2}, \mathbf{u}_{1} \perp \mathbf{u}_{4}$, and $\mathbf{u}_{2} \perp \mathbf{u}_{3}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.4.12
|
Compute $A^{4}$, where
$$
A=\left[\begin{array}{ll}
1 & -3 \\
2 & -5
\end{array}\right]
$$
Hint: you can save some work by calculating $A^{2}$ times $A^{2}$.
|
$\left[\begin{array}{rr}-71 & 168 \\ -112 & 265\end{array}\right]$
|
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|
college_math.linear_algebra
|
\left[\begin{array}{rr}-71 & 168 \\ -112 & 265\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.11.9
|
Unitarily diagonalize the hermitian matrix $A=\left[\begin{array}{cc}2 & 1+i \\ 1-i & 3\end{array}\right]$.
|
Eigenvalues: $\lambda_{1}=1$ and $\lambda_{2}=4$. Normalized eigenvectors: $\mathbf{v}_{1}=\frac{1}{\sqrt{3}}\left[\begin{array}{c}1+i \\ -1\end{array}\right], \mathbf{v}_{2}=\frac{1}{\sqrt{3}}\left[\begin{array}{c}1 \\ 1-i\end{array}\right]$.
$$
D=\left[\begin{array}{ll}
1 & 0 \\
0 & 4
\end{array}\right], \quad P=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}
1+i & 1 \\
-1 & 1-i
\end{array}\right] \text {. }
$$`
|
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|
college_math.linear_algebra
|
Eigenvalues: $\lambda_{1}=1$ and $\lambda_{2}=4$. Normalized eigenvectors: $\mathbf{v}_{1}=\frac{1}{\sqrt{3}}\left[\begin{array}{c}1+i \\ -1\end{array}\right], \mathbf{v}_{2}=\frac{1}{\sqrt{3}}\left[\begin{array}{c}1 \\ 1-i\end{array}\right]$.
$$
D=\left[\begin{array}{ll}
1 & 0 \\
0 & 4
\end{array}\right], \quad P=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}
1+i & 1 \\
-1 & 1-i
\end{array}\right] \text {. }
$$`
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.5.4
|
Consider the points $\left(x_{1}, y_{1}\right)=(-1,4),\left(x_{2}, y_{2}\right)=(0,-2),\left(x_{3}, y_{3}\right)=(1,4),\left(x_{4}, y_{4}\right)=$ $(2,2)$. Find the least squares parabola for these points.
|
$y=1-x+x^{2}$.
|
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|
college_math.linear_algebra
|
y=1-x+x^{2}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.7.5
|
Find the area of the parallelogram with vertices $(-2,3,1),(2,1,1),(1,2,-1)$, and $(5,0,-1)$.
|
Let $P=(-2,3,1), Q=(2,1,1), R=(1,2,-1)$, and $S=(5,0,-1)$. We have $\overrightarrow{P Q} \times \overrightarrow{P R}=\left[\begin{array}{r}4 \\ -2 \\ 0\end{array}\right] \times$ $\left[\begin{array}{r}3 \\ -1 \\ -2\end{array}\right]=\left[\begin{array}{l}5 \\ 8 \\ 2\end{array}\right]$. The area of the parallelogram is $\|\overrightarrow{P Q} \times \overrightarrow{P R}\|=\sqrt{5^{2}+8^{2}+2^{2}}=\sqrt{93}$
|
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college_math.linear_algebra
|
Let $P=(-2,3,1), Q=(2,1,1), R=(1,2,-1)$, and $S=(5,0,-1)$. We have $\overrightarrow{P Q} \times \overrightarrow{P R}=\left[\begin{array}{r}4 \\ -2 \\ 0\end{array}\right] \times$ $\left[\begin{array}{r}3 \\ -1 \\ -2\end{array}\right]=\left[\begin{array}{l}5 \\ 8 \\ 2\end{array}\right]$. The area of the parallelogram is $\|\overrightarrow{P Q} \times \overrightarrow{P R}\|=\sqrt{5^{2}+8^{2}+2^{2}}=\sqrt{93}
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.9.2.9
|
Assume $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are linearly independent elements of some vector space $V$. Consider the set of vectors
$$
R=\left\{2 \mathbf{u}-\mathbf{w}, \mathbf{w}+\mathbf{v}, 3 \mathbf{v}+\frac{1}{2} \mathbf{u}\right\}
$$
Determine whether $R$ is linearly independent.
|
Therefore, the set $R$ is linearly independent.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Therefore, the set $R$ is linearly independent.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.5.2.1
|
Which of the following vectors are redundant? If there are redundant vectors, write each of them as a linear combination of previous vectors.
$$
\mathbf{u}_{1}=\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{l}
2 \\
0 \\
2
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{l}
1 \\
2 \\
1
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{l}
1 \\
6 \\
1
\end{array}\right]
$$
|
$\mathbf{u}_{2}$ and $\mathbf{u}_{4}$ are redundant. We have $\mathbf{u}_{2}=2 \mathbf{u}_{1}$ and $\mathbf{u}_{4}=3 \mathbf{u}_{3}-2 \mathbf{u}_{1}$.
|
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|
college_math.linear_algebra
|
\mathbf{u}_{2}$ and $\mathbf{u}_{4}$ are redundant. We have $\mathbf{u}_{2}=2 \mathbf{u}_{1}$ and $\mathbf{u}_{4}=3 \mathbf{u}_{3}-2 \mathbf{u}_{1}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.5.2.6
|
Find a linearly independent set of vectors that has the same span as the given vectors.
$$
\mathbf{u}_{1}=\left[\begin{array}{l}
2 \\
0 \\
3
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{l}
1 \\
3 \\
5
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{l}
3 \\
3 \\
8
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{r}
3 \\
-3 \\
1
\end{array}\right]
$$
|
$\left[\begin{array}{rrrr}2 & 1 & 3 & 3 \\ 0 & 3 & 3 & -3 \\ 3 & 5 & 8 & 1\end{array}\right] \simeq\left[\begin{array}{rrrr}1 & 4 & 5 & -2 \\ 0 & 1 & 1 & -1 \\ 0 & 0 & 0 & 0\end{array}\right]$. Linearly independent subset: $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$.
|
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|
college_math.linear_algebra
|
\left[\begin{array}{rrrr}2 & 1 & 3 & 3 \\ 0 & 3 & 3 & -3 \\ 3 & 5 & 8 & 1\end{array}\right] \simeq\left[\begin{array}{rrrr}1 & 4 & 5 & -2 \\ 0 & 1 & 1 & -1 \\ 0 & 0 & 0 & 0\end{array}\right]$. Linearly independent subset: $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.5.8
|
Which of the following are unit vectors?
$$
\mathbf{u}=\frac{1}{2}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right], \quad \mathbf{v}=\frac{1}{3}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right], \quad \mathbf{w}=\frac{1}{2}\left[\begin{array}{l}
1 \\
1 \\
1 \\
1
\end{array}\right]
$$
|
Only $\mathbf{w}$ is a unit vector.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Only $\mathbf{w}$ is a unit vector.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.6.9
|
Find $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$ where $\mathbf{w}=\left[\begin{array}{r}1 \\ 2 \\ -2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]$.
|
$\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}=\frac{-5}{10}\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]=\left[\begin{array}{c}-\frac{1}{2} \\ 0 \\ -\frac{3}{2}\end{array}\right]$
|
Creative Commons Attribution License (CC BY)
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college_math.linear_algebra
|
\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}=\frac{-5}{10}\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]=\left[\begin{array}{c}-\frac{1}{2} \\ 0 \\ -\frac{3}{2}\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.6.14
|
Decompose the vector $\mathbf{v}$ into $\mathbf{v}=\mathbf{a}+\mathbf{b}$ where $\mathbf{a}$ is parallel to $\mathbf{u}$ and $\mathbf{b}$ is orthogonal to $\mathbf{u}$.
$$
\mathbf{v}=\left[\begin{array}{r}
3 \\
2 \\
-5
\end{array}\right], \quad \mathbf{u}=\left[\begin{array}{r}
1 \\
-1 \\
2
\end{array}\right]
$$
|
$$
\mathbf{a}=\left[\begin{array}{c}
-3 / 2 \\
3 / 2 \\
-3
\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c}
9 / 2 \\
1 / 2 \\
-2
\end{array}\right]
$$
|
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|
college_math.linear_algebra
|
\mathbf{a}=\left[\begin{array}{c}
-3 / 2 \\
3 / 2 \\
-3
\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c}
9 / 2 \\
1 / 2 \\
-2
\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.1.1
|
Can a column vector ever be equal to a row vector?
|
Yes, a $1 \times 1$-matrix is both a row vector and a column vector. However, a column vector of dimension 2 or greater can never be equal to a row vector, because one is an $n \times 1$-matrix and the other is a $1 \times n$ matrix.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Yes, a $1 \times 1$-matrix is both a row vector and a column vector. However, a column vector of dimension 2 or greater can never be equal to a row vector, because one is an $n \times 1$-matrix and the other is a $1 \times n$ matrix.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.3.1
|
In $\mathbb{R}^{3}$ with the usual dot product, find an orthogonal basis for
$$
\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right],\left[\begin{array}{l}
2 \\
6 \\
0
\end{array}\right]\right\}
$$
|
$\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 4 \\ -3\end{array}\right]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 4 \\ -3\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.A.1.1
|
Let $z=2+7 i$ and let $w=3-8 i$. Compute $z+w, z-2 w$, $z w$, and $\frac{w}{z}$.
|
$z+w=5-i, z-2 w=-4+23 i, z w=62+5 i$, and $\frac{w}{z}=-\frac{50}{53}-\frac{37}{53} i$.
|
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|
college_math.linear_algebra
|
z+w=5-i, z-2 w=-4+23 i, z w=62+5 i$, and $\frac{w}{z}=-\frac{50}{53}-\frac{37}{53} i$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.4.8
|
Find $h$ such that
$$
\left[\begin{array}{ll|l}
1 & h & 3 \\
2 & 4 & 6
\end{array}\right]
$$
is the augmented matrix of a consistent system.
|
Any $h$ will work.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Any $h$ will work.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.3.2
|
Use elementary operations to find the point $(x, y)$ that lies on both lines $x+3 y=1$ and $4 x-y=3$.
|
Solution is: $(x, y)=\left(\frac{10}{13}, \frac{1}{13}\right)$
|
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|
college_math.linear_algebra
|
Solution is: $(x, y)=\left(\frac{10}{13}, \frac{1}{13}\right)
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.5.1.3
|
Describe the span of the following vectors in $\mathbb{R}^{4}$:
$$
\mathbf{u}_{1}=\left[\begin{array}{r}
1 \\
1 \\
-1 \\
-1
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{l}
5 \\
1 \\
1 \\
0
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{r}
0 \\
-2 \\
2 \\
1
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{l}
3 \\
1 \\
1 \\
1
\end{array}\right]
$$
|
It is the hyperplane given by the equation $x-2 y-3 z+2 w=0$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
It is the hyperplane given by the equation $x-2 y-3 z+2 w=0$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.2.1
|
Find $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]+\left[\begin{array}{l}1 \\ 5 \\ 1\end{array}\right]+\left[\begin{array}{c}-1 \\ 2 \\ -4\end{array}\right]$.
|
$\left[\begin{array}{l}1 \\ 9 \\ 0\end{array}\right]$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\left[\begin{array}{l}1 \\ 9 \\ 0\end{array}\right]$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.6.3
|
My system of equations has a solution $(x, y, z)=(1,2,4)$. The associated homogeneous system has basic solutions $(x, y, z)=(1,0,1)$ and $(x, y, z)=(0,1,-1)$. What is the general solution of my system of equations?
|
The general solution is $(x, y, z)=(1,2,4)+s(1,0,1)+t(0,1,-1)$, or equivalently, $(x, y, z)=(1+$ $s, 2+t, 4+s-t)$, where $s$ and $t$ are parameters.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
The general solution is $(x, y, z)=(1,2,4)+s(1,0,1)+t(0,1,-1)$, or equivalently, $(x, y, z)=(1+$ $s, 2+t, 4+s-t)$, where $s$ and $t$ are parameters.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.8.4.3
|
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
5 & -18 & -32 \\
0 & 5 & 4 \\
2 & -5 & -11
\end{array}\right]
$$
One eigenvalue is 1. Diagonalize if possible.
|
The eigenvalues are -1 and 1. The eigenvectors corresponding to the eigenvalues are:
$$
\left\{\left[\begin{array}{c}
10 \\
-2 \\
3
\end{array}\right]\right\} \text { for eigenvalue }-1, \quad\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \text { for eigenvalue } 1 .
$$
Since there are only 2 linearly independent eigenvectors, this matrix is not diagonalizable.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
The eigenvalues are -1 and 1. The eigenvectors corresponding to the eigenvalues are:
$$
\left\{\left[\begin{array}{c}
10 \\
-2 \\
3
\end{array}\right]\right\} \text { for eigenvalue }-1, \quad\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \text { for eigenvalue } 1 .
$$
Since there are only 2 linearly independent eigenvectors, this matrix is not diagonalizable.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.5.5
|
Find the length of each of the following vectors.
$$
\mathbf{u}=\left[\begin{array}{c}
-3 \\
2
\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{c}
1 \\
-2 \\
5
\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{c}
1 \\
4 \\
-2 \\
1
\end{array}\right]
$$
|
$\|\mathbf{u}\|=\sqrt{13},\|\mathbf{v}\|=\sqrt{30},\|\mathbf{w}\|=\sqrt{22}$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\|\mathbf{u}\|=\sqrt{13},\|\mathbf{v}\|=\sqrt{30},\|\mathbf{w}\|=\sqrt{22}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.4.9
|
Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 2 \\ 3 & k\end{array}\right]$. Is it possible to find $k$ such that $A B=B A$? If so, what should $k$ equal?
|
Solution is: $k=4$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Solution is: $k=4$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.9.2
|
Decrypt the message "ERM DXYBJUWW. JWQLD, HL" using the Hill cipher with block size 3 and encryption matrix
$$
A=\left[\begin{array}{lll}
2 & 1 & 1 \\
1 & 3 & 1 \\
1 & 1 & 4
\end{array}\right]
$$
|
The decryption matrix is
$$
A^{-1}=\left[\begin{array}{ccc}
16 & 22 & 5 \\
22 & 26 & 17 \\
5 & 17 & 2
\end{array}\right]
$$
Plaintext: "Spies are at the gate".
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
The decryption matrix is
$$
A^{-1}=\left[\begin{array}{ccc}
16 & 22 & 5 \\
22 & 26 & 17 \\
5 & 17 & 2
\end{array}\right]
$$
Plaintext: "Spies are at the gate".
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.3.2
|
Find $-3\left[\begin{array}{r}5 \\ -1 \\ 2 \\ -3\end{array}\right]+5\left[\begin{array}{r}-8 \\ 2 \\ -3 \\ 6\end{array}\right]$.
|
$\left[\begin{array}{r}-55 \\ 13 \\ -21 \\ 39\end{array}\right]$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\left[\begin{array}{r}-55 \\ 13 \\ -21 \\ 39\end{array}\right]$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.3.4
|
Let
$$
A=\left[\begin{array}{ccc}
3 & -1 & 0 \\
-1 & 5 & 2 \\
0 & 2 & 3
\end{array}\right]
$$
and consider the vector space $\mathbb{R}^{3}$ with the inner product given by $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T}$ Aw. Let
$$
\mathbf{v}_{1}=\left[\begin{array}{l}
1 \\
0 \\
2
\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{c}
-1 \\
1 \\
-5
\end{array}\right], \quad \text { and } \quad \mathbf{v}_{3}=\left[\begin{array}{l}
2 \\
2 \\
3
\end{array}\right]
$$
Apply the Gram-Schmidt procedure to $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$ to find an orthogonal basis $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ for $\mathbb{R}^{3}$ with respect to the above inner product.
|
$\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 0 \\ 2\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{c}-1 \\ 1 \\ 0\end{array}\right]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 0 \\ 2\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{c}-1 \\ 1 \\ 0\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.9.3
|
Eve intercepts the following encrypted message sent by Bob:
“TGVXKHGSW, JU, JHY JSCDSBQIRPEV”
Eve knows that Alice uses a Hill cipher with block length 2, but she does not know the secret encryption matrix. Eve also knows that Bob begins all of his letters with "Hello". Decrypt the message.
|
The first two plaintext blocks are $(8,5),(12,12)$ and the first two ciphertext blocks are $(20,7),(22,24)$. Eve solves the equation
$$
A^{-1}\left[\begin{array}{cc}
20 & 22 \\
7 & 24
\end{array}\right]=\left[\begin{array}{ll}
8 & 12 \\
5 & 12
\end{array}\right]
$$
to find the secret decryption matrix
$$
A^{-1}=\left[\begin{array}{ll}
3 & 5 \\
1 & 2
\end{array}\right] .
$$
The plaintext is "Hello, password is kiwifruit".
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
The first two plaintext blocks are $(8,5),(12,12)$ and the first two ciphertext blocks are $(20,7),(22,24)$. Eve solves the equation
$$
A^{-1}\left[\begin{array}{cc}
20 & 22 \\
7 & 24
\end{array}\right]=\left[\begin{array}{ll}
8 & 12 \\
5 & 12
\end{array}\right]
$$
to find the secret decryption matrix
$$
A^{-1}=\left[\begin{array}{ll}
3 & 5 \\
1 & 2
\end{array}\right] .
$$
The plaintext is "Hello, password is kiwifruit".
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.2.5
|
Suppose $B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ is an orthogonal basis for an inner product space $V$, such that $\left\|\mathbf{u}_{1}\right\|=2,\left\|\mathbf{u}_{2}\right\|=\sqrt{3}$, and $\left\|\mathbf{u}_{3}\right\|=\sqrt{5}$. Moreover, suppose that $\mathbf{v} \in V$ is a vector such that $\left\langle\mathbf{v}, \mathbf{u}_{1}\right\rangle=1$, $\left\langle\mathbf{v}, \mathbf{u}_{2}\right\rangle=2$, and $\left\langle\mathbf{v}, \mathbf{u}_{3}\right\rangle=-4$. Find the coordinates of $\mathbf{v}$ with respect to $B$.
|
$\mathbf{v}=\frac{1}{4} \mathbf{u}_{1}+\frac{2}{3} \mathbf{u}_{2}-\frac{4}{5} \mathbf{u}_{3}$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\mathbf{v}=\frac{1}{4} \mathbf{u}_{1}+\frac{2}{3} \mathbf{u}_{2}-\frac{4}{5} \mathbf{u}_{3}
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.7.12
|
Find the volume of the parallelepiped determined by the vectors $\left[\begin{array}{r}1 \\ -7 \\ -5\end{array}\right],\left[\begin{array}{r}1 \\ -2 \\ -6\end{array}\right]$, and $\left[\begin{array}{l}3 \\ 2 \\ 3\end{array}\right]$
|
\left(\left[\begin{array}{l}1 \\ 1 \\ 3\end{array}\right] \times\left[\begin{array}{r}-7 \\ -2 \\ 2\end{array}\right]\right) \cdot\left[\begin{array}{r}-5 \\ -6 \\ 3\end{array}\right]=\left[\begin{array}{r}8 \\ -23 \\ 5\end{array}\right] \cdot\left[\begin{array}{r}-5 \\ -6 \\ 3\end{array}\right]=113$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\left(\left[\begin{array}{l}1 \\ 1 \\ 3\end{array}\right] \times\left[\begin{array}{r}-7 \\ -2 \\ 2\end{array}\right]\right) \cdot\left[\begin{array}{r}-5 \\ -6 \\ 3\end{array}\right]=\left[\begin{array}{r}8 \\ -23 \\ 5\end{array}\right] \cdot\left[\begin{array}{r}-5 \\ -6 \\ 3\end{array}\right]=113
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.3.3
|
In $\mathbb{R}^{4}$ with the usual dot product, find an orthogonal basis for
$$
\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
0 \\
1 \\
0
\end{array}\right],\left[\begin{array}{c}
1 \\
3 \\
1 \\
-1
\end{array}\right],\left[\begin{array}{l}
2 \\
4 \\
2 \\
2
\end{array}\right]\right\}
$$
|
$\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}0 \\ 3 \\ 0 \\ -1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{l}0 \\ 1 \\ 0 \\ 3\end{array}\right]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}0 \\ 3 \\ 0 \\ -1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{l}0 \\ 1 \\ 0 \\ 3\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.7.6.8
|
Consider the matrix
$$
A=\left[\begin{array}{ccc}
e^{t} & \cosh t & \sinh t \\
e^{t} & \sinh t & \cosh t \\
e^{t} & \cosh t & \sinh t
\end{array}\right]
$$
Does there exist a value of t for which this matrix fails to be invertible? Explain.
|
Since the matrix $A$ has two identical rows, we have $\operatorname{det}(A)=0$ for all $t$. So this matrix is noninvertible for all $t$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Since the matrix $A$ has two identical rows, we have $\operatorname{det}(A)=0$ for all $t$. So this matrix is noninvertible for all $t$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.3.6
|
Consider the inner product space $C[0,2]$, with the inner product given by
$$
\langle p, q\rangle=\int_{0}^{2} f(x) g(x) d x .
$$
Use the Gram-Schmidt procedure to find an orthogonal basis for $\operatorname{span}\left\{1, x, x^{2}\right\}$.
|
$\mathbf{u}_{1}=1, \mathbf{u}_{2}=x-1, x^{2}-2 x+\frac{2}{3}$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\mathbf{u}_{1}=1, \mathbf{u}_{2}=x-1, x^{2}-2 x+\frac{2}{3}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.4.2
|
Decide whether
$$
\mathbf{v}=\left[\begin{array}{r}
4 \\
4 \\
-3
\end{array}\right]
$$
is a linear combination of the vectors
$$
\mathbf{u}_{1}=\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right] \quad \text { and } \quad \mathbf{u}_{2}=\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$
If yes, find the coefficients.
|
$$
\left[\begin{array}{r}
4 \\
4 \\
-3
\end{array}\right]=2\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right]-\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\left[\begin{array}{r}
4 \\
4 \\
-3
\end{array}\right]=2\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right]-\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.6.8
|
Find $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$ where $\mathbf{w}=\left[\begin{array}{r}1 \\ 0 \\ -2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$.
|
$\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}=\frac{-5}{14}\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]=\left[\begin{array}{c}-\frac{5}{14} \\ -\frac{5}{7} \\ -\frac{15}{14}\end{array}\right]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}=\frac{-5}{14}\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]=\left[\begin{array}{c}-\frac{5}{14} \\ -\frac{5}{7} \\ -\frac{15}{14}\end{array}\right]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.4.2.3
|
Let $A=\left[\begin{array}{rrr}1 & 2 & -1 \\ -1 & 4 & 0\end{array}\right]$ and $B=\left[\begin{array}{rrr}0 & 3 & 0 \\ 1 & -1 & 1\end{array}\right]$.
Find a matrix $X$ such that $(A+X)-(B+0)=B+A$. Hint: first use the properties of matrix addition to simplify the equation and solve for $X$.
|
The equation simplifies to $X=B+B$, so $X=\left[\begin{array}{rrr}0 & 6 & 0 \\ 2 & -2 & 2\end{array}\right]$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
The equation simplifies to $X=B+B$, so $X=\left[\begin{array}{rrr}0 & 6 & 0 \\ 2 & -2 & 2\end{array}\right]$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.5.4
|
Use Gauss-Jordan elimination to solve the system of equations $3 x-y-2 z=3, y-4 z=0$, and $-2 x+y=-2$.
|
Solution is: $[x=2 t+1, y=4 t, z=t]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Solution is: $[x=2 t+1, y=4 t, z=t]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.6.4
|
Find $\cos \theta$ where $\theta$ is the angle between the vectors
$$
\mathbf{u}=\left[\begin{array}{r}
3 \\
-1 \\
-1
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
4 \\
2
\end{array}\right]
$$
|
$\cos \theta=\frac{\left[\begin{array}{ccc}3 & -1 & -1\end{array}\right]^{T} \cdot\left[\begin{array}{lll}1 & 4 & 2\end{array}\right]^{T}}{\sqrt{9+1+1} \sqrt{1+16+4}}=\frac{-3}{\sqrt{11} \sqrt{21}}$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\cos \theta=\frac{\left[\begin{array}{ccc}3 & -1 & -1\end{array}\right]^{T} \cdot\left[\begin{array}{lll}1 & 4 & 2\end{array}\right]^{T}}{\sqrt{9+1+1} \sqrt{1+16+4}}=\frac{-3}{\sqrt{11} \sqrt{21}}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.2.7.6
|
Find the area of the triangle determined by the three points, $(1,0,3),(4,1,0)$ and $(-3,1,1)$.
|
Let $P=(1,0,3), Q=(4,1,0)$, and $R=(-3,1,1) . \overrightarrow{P Q} \times \overrightarrow{P R}=\left[\begin{array}{r}3 \\ 1 \\ -3\end{array}\right] \times\left[\begin{array}{r}-4 \\ 1 \\ -2\end{array}\right]=\left[\begin{array}{c}1 \\ 18 \\ 7\end{array}\right]$. The area of the triangle is $\frac{1}{2}\|\overrightarrow{P Q} \times \overrightarrow{P R}\|=\frac{1}{2} \sqrt{1+18^{2}+7^{2}}=\frac{1}{2} \sqrt{374}$.
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Let $P=(1,0,3), Q=(4,1,0)$, and $R=(-3,1,1) . \overrightarrow{P Q} \times \overrightarrow{P R}=\left[\begin{array}{r}3 \\ 1 \\ -3\end{array}\right] \times\left[\begin{array}{r}-4 \\ 1 \\ -2\end{array}\right]=\left[\begin{array}{c}1 \\ 18 \\ 7\end{array}\right]$. The area of the triangle is $\frac{1}{2}\|\overrightarrow{P Q} \times \overrightarrow{P R}\|=\frac{1}{2} \sqrt{1+18^{2}+7^{2}}=\frac{1}{2} \sqrt{374}$.
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.1.5.5
|
Use Gauss-Jordan elimination to solve the system of equations $-9 x+15 y=66,-11 x+$ $18 y=79,-x+y=4$, and $z=3$.
|
Solution is: $[x=1, y=5, z=3]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
Solution is: $[x=1, y=5, z=3]
|
college_math.Matrix_Theory_and_Linear_Algebra
|
exercise.11.3.5
|
Let
$$
A=\left[\begin{array}{llll}
3 & 2 & 0 & 0 \\
2 & 5 & 1 & 0 \\
0 & 1 & 3 & 1 \\
0 & 0 & 1 & 3
\end{array}\right]
$$
and consider the vector space $\mathbb{R}^{4}$ with the inner product given by $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T}$ Aw. Let
$$
\mathbf{v}_{1}=\left[\begin{array}{c}
1 \\
-1 \\
1 \\
1
\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{l}
2 \\
0 \\
0 \\
2
\end{array}\right], \quad \text { and } \quad \mathbf{v}_{3}=\left[\begin{array}{l}
2 \\
0 \\
2 \\
3
\end{array}\right]
$$
and let $W=\operatorname{span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$. Apply the Gram-Schmidt procedure to $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$ to find an orthogonal basis $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ for $W$ with respect to the above inner product.
|
$\mathbf{u}_{1}=\left[\begin{array}{c}1 \\ -1 \\ 1 \\ 1\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 1 \\ -1 \\ 1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{c}-1 \\ 1 \\ 1 \\ 0\end{array}\right]$
|
Creative Commons Attribution License (CC BY)
|
college_math.linear_algebra
|
\mathbf{u}_{1}=\left[\begin{array}{c}1 \\ -1 \\ 1 \\ 1\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 1 \\ -1 \\ 1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{c}-1 \\ 1 \\ 1 \\ 0\end{array}\right]
|
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