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Question: <p>Is there a connection among norm of a matrix, its eigenvalues and the image of <span class="math-container">$A$</span>? Specifically if all eigenvalues of a matrix <span class="math-container">$A$</span> (<span class="math-container">$n$</span> by <span class="math-container">$n$</span>) have absolute valu... | https://math.stackexchange.com/questions/3721304/connection-between-eigenvalues-of-a-real-matrix-a-and-its-norm |
Question: <p>Let <span class="math-container">$F$</span> be a finite field with <span class="math-container">$q$</span> elements. Show that every function <span class="math-container">$f \in \mathcal{F}(F,F)$</span> is uniquely a polynomial of degree <span class="math-container">$\leq q−1$</span> with coefficients in <... | https://math.stackexchange.com/questions/3722183/show-that-every-function-f-in-mathcalff-f-is-uniquely-a-polynomial-of-d |
Question: <p>I am counting something wrongly as I am looking at the determinant of a block matrix.</p>
<p>Let us consider this example:
<span class="math-container">$$
M = \begin{pmatrix} a & 1^T \\ 1 & I_{n-1}\end{pmatrix}
$$</span>
where <span class="math-container">$a\neq 0$</span> is scalar, <span class="... | https://math.stackexchange.com/questions/3722339/something-is-counted-wrongly-in-a-determinant-of-block-matrices |
Question: <p>Consider the linear space $c^{(3)}$ of all sequences $x = (x_n)_{n=1}^{\infty}$ such that $\{x_{3k+q} \}_{k=0}^{\infty}$ converges for $q = 0,1,2$. Find the dimension and a basis for $c^{(3)}/c_0$. Note that $c_0$ is the linear space of sequences that converge to $0$.</p>
<p>I think the dimension is $1$ u... | https://math.stackexchange.com/questions/16490/dimension-and-sequences |
Question: <p>A subspace V in $R^4$ is defined by the equation $x_{1}$-$x_{2}$+$2x_{3}$+$4x_{4}$=0. I need to find T such that Ker(T)=zero vector, and Im(T)=V. How do I approach this problem? As I understand, the equation given to me is a set of points that are solutions to Im(T)=V, so in a sense they vectors of that pl... | https://math.stackexchange.com/questions/19335/find-a-t-from-r3-to-r4-given-an-equation-for-a-subspace-in-r4 |
Question: <p>I'm confronted with this question:</p>
<blockquote>
<p>Let <span class="math-container">$V$</span> be an inner product space and <span class="math-container">$B=\{u_{1}, ..., u_{n}\}$</span> a basis of <span class="math-container">$V$</span>.</p>
<p>Suppose there exists <span class="math-container">$\lambd... | https://math.stackexchange.com/questions/26779/regarding-orthonormal-basis |
Question: <p>How can I prove that a 3x3 system of linear equations of the form:</p>
<p>$\begin{pmatrix}
a&a+b&a+2b\\
c&c+d&c+2d\\
e&e+f&e+2f
\end{pmatrix}
\begin{pmatrix}
x\\ y\\ z
\end{pmatrix}
=\begin{pmatrix}
a+3b\\
c+3d\\
e+3f
\end{pmatrix}$</p>
<p>for $a,b,c,d,e,f \in \mathbb ... | https://math.stackexchange.com/questions/27259/prove-a-3x3-system-of-linear-equations-with-arithmetic-progression-coefficients |
Question: <p>Given an invertible $3\times 3$ matrix:</p>
<p>$A = \begin{pmatrix}
1 & 2 & 2 \\
1 & 2 & -1 \\
-1 & 1 & 4
\end{pmatrix}$</p>
<p>I am trying to find $f(x)$ from $F[x]$ such that $A^{-1}=f(A)$. To do so, I want to use the result of <a href="https://math.stackexchange.com/questio... | https://math.stackexchange.com/questions/29580/trying-to-find-fx-in-fx-such-that-fa-a-1 |
Question: <blockquote>
<p>Let <span class="math-container">$T: \mathbb{R}^{3} \to \mathbb{R}^{3}$</span> be the following linear operator, which rotates each vector <span class="math-container">$v$</span> about the <span class="math-container">$z$</span>-axis by an angle <span class="math-container">$\theta$</span>: <s... | https://math.stackexchange.com/questions/49267/help-understanding-this-example-of-a-linear-operator-which-rotates-each-vector |
Question: <p>Given a real square matrix <span class="math-container">$A$</span>, we can factor it as <span class="math-container">$$A = QR$$</span> where <span class="math-container">$Q$</span> is orthogonal and <span class="math-container">$R$</span> is upper triangular. The entries of <span class="math-container">$R$... | https://math.stackexchange.com/questions/67803/on-a-matrix-factorization-and-the-gram-schmidt-process |
Question: <blockquote>
<p>We're given <span class="math-container">$V$</span>, which is an <span class="math-container">$n$</span> dimensional vector space. <span class="math-container">$T : V \to V$</span> is a linear transformation. There is a vector <span class="math-container">$v \in V$</span> such that <span class... | https://math.stackexchange.com/questions/91324/finding-the-matrix-of-this-linear-transformation |
Question: <p>I came along with the following exercise that I developed poorly. May anybody give me some light? See:</p>
<blockquote>
<p>How to find a solution involving <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> to make the following syste... | https://math.stackexchange.com/questions/213580/find-values-for-a-b-c-that-make-this-linear-system-solvable |
Question: <blockquote>
<p>13. Suppose <span class="math-container">$V$</span> and <span class="math-container">$W$</span> are finite-dimensional vector spaces and <span class="math-container">$T:V \to W$</span> is an isomorphism. Then there exist bases <span class="math-container">$\mathcal{B}$</span> and <span class="... | https://math.stackexchange.com/questions/255298/linear-algebra-linear-transformations |
Question: <blockquote>
<p>If A is a <span class="math-container">$m\times n$</span> matrix and <span class="math-container">$M = (A \mid b)$</span> the augmented matrix
for the linear system <span class="math-container">$Ax = b$</span>.</p>
<p>Show that either<br> <br><span class="math-container">$(i) \operatorname{ran... | https://math.stackexchange.com/questions/255898/linear-algebra-proof |
Question: <p>The question is:</p>
<blockquote>
<p>Give a formal proof for the following statement:
Given a matrix A and a scalar c, show that rank(cA) = rank(A)</p>
</blockquote>
<p>Here are the steps that I took to go about the proof:</p>
<blockquote>
<p>(1) Prove this claim: Let v1, v2, ..., vN be vectors</p>
<p>then... | https://math.stackexchange.com/questions/301629/question-on-a-proof-about-the-rank-of-a-matrix |
Question: <blockquote>
<p>Let <span class="math-container">$V$</span> be a finite dimensional vector space over a field <span class="math-container">$F$</span>.</p>
<p>Let <span class="math-container">$v\in V$</span> with <span class="math-container">$v$</span> not equal to <span class="math-container">$0$</span>. Show... | https://math.stackexchange.com/questions/354235/dual-space-questions |
Question: <blockquote>
<p><span class="math-container">$\newcommand{\sp}{\operatorname{sp}}$</span> Let <span class="math-container">$V$</span> be a vector space over <span class="math-container">$F$</span> field, and let <span class="math-container">$A,B$</span> be two different, disjoint, non-empty sets of vectors fr... | https://math.stackexchange.com/questions/392636/if-the-union-of-a-and-b-is-linearly-independent-then-the-intersection-of-the |
Question: <p>Definition: Let <span class="math-container">$V$</span> be vector space, and <span class="math-container">$U$</span>, <span class="math-container">$W$</span> be two subspaces such that <span class="math-container">$V=U\oplus W$</span>.</p>
<p>We know that there exists for each <span class="math-container">... | https://math.stackexchange.com/questions/397065/projection-and-inner-product-space |
Question: <blockquote>
<p>a. find <span class="math-container">$||f||$</span></p>
<p>b. find all linear polynomials that are orthogonal to <span class="math-container">$x$</span></p>
</blockquote>
<p>Okay, so I know that</p>
<p><span class="math-container">$||f|| = \sqrt(f_1^2 + f_2^2 +... + f_n^2)$</span></p>
<p>and t... | https://math.stackexchange.com/questions/561785/on-c0-0-1-define-f-cdot-g-int-01-fx-gx-dx-for-fx-x |
Question: <blockquote>
<p><span class="math-container">$A$</span> and <span class="math-container">$B$</span> are matrices of order <span class="math-container">$m\times n$</span>.</p>
<p>Prove or disprove: If <span class="math-container">$Null(A-B)=\mathbb R^n$</span> then <span class="math-container">$ A=B $</span></... | https://math.stackexchange.com/questions/617541/prove-or-disprove-if-nulla-b-mathbb-rn-then-a-b |
Question: <p>We've given that <span class="math-container">$V$</span> is a vector space and that <span class="math-container">$L(V)$</span> the set with functions <span class="math-container">$T:V\rightarrow \mathbb{R}$</span> s.t. <span class="math-container">$T(a_1f_1+a_2f_2)=a_1T(f_1)+a_2T(f_2)$</span>. We must show... | https://math.stackexchange.com/questions/984816/proving-that-a-set-of-functions-is-a-vector-space |
Question: <p>Given K,L are sub-sets of <span class="math-container">$K^4$</span>:</p>
<p><span class="math-container">$K = \{(-5,8,14,0),(-1,4,2,4)\}, L = \{(0,1,-10,8),(0,3,-1,5)\}$</span></p>
<blockquote>
<p>Find a homogeneous system of equations that its solutions are Spanned by K.</p>
<p>Also prove that L spans the... | https://math.stackexchange.com/questions/852442/homogeneous-system-of-equations-and-sub-set-k-of-r4 |
Question: <blockquote>
<p>Let <span class="math-container">$A,B$</span> be <span class="math-container">$n\times n$</span> matrices.</p>
<p>If <span class="math-container">$AB=0$</span>, prove that the columns of matrix <span class="math-container">$B$</span> are vectors in the kernel of <span class="math-container">$A... | https://math.stackexchange.com/questions/1095451/if-ab-0-prove-that-the-columns-of-matrix-b-are-vectors-in-the-kernel-of |
Question: <blockquote>
<p>Choose the correct set of functions, which are not linearly independent.</p>
<ol>
<li><span class="math-container">$x^2-1$</span>, <span class="math-container">$2x^2-x+1$</span>, <span class="math-container">$3x^2-x$</span></li>
<li><span class="math-container">$1$</span>, <span class="math-co... | https://math.stackexchange.com/questions/1097550/how-do-i-find-which-set-of-functions-is-linearly-independent |
Question: <p>Suppose</p>
<blockquote>
<p><span class="math-container">$A$</span> is our matrix</p>
<p><span class="math-container">$B$</span> is our basis for the matrix <span class="math-container">$A$</span></p>
<p><span class="math-container">$Q$</span> is orthogonal basis for matrix <span class="math-container">$A$... | https://math.stackexchange.com/questions/1303925/projections-onto-a-subspace-orthogonal-vs-non-orthogonal-matrix-vs-basis-matr |
Question: <p>Hello I'm trying to solve this question from Ron Larson's linear algebra textbook. But I'm just stuck on how to approach this question. Could someone please at least give me a hint on how to approach this sort of question?</p>
<blockquote>
<p>Suppose <span class="math-container">$T:\unicode{x211D}^2\righta... | https://math.stackexchange.com/questions/779180/understanding-a-linear-transformation |
Question: <p>Let $V$ and $W$ be vector spaces with $\dim V = \dim W$. If $T : V → W$ is
linear then $T$ is one-to-one if and only if $T$ is onto.
But this is true only when the dimensions of $V$ and $W$ are finite.
For instance I came across the example $T : P(R)\to P(R)$ such that $T(f(x))=f'(x)$. Here T is onto but ... | https://math.stackexchange.com/questions/1198165/dimension-theorem-corollary |
Question: <p><span class="math-container">$${\color{brown}{\text{Question I am trying to solve:}}}$$</span></p>
<p>Let <span class="math-container">$A,B$</span> and <span class="math-container">$X$</span> be 7 x 7 matrices such that <span class="math-container">$\det A=1$</span>, <span class="math-container">$\det B=3$... | https://math.stackexchange.com/questions/1332815/does-abc-d-implies-detabc-detd |
Question: <p>I have an exercise to answer, and I don't know if I've done it the right way. This is only a little part of the exercise, but I have to know if what I've done so far is correct. Here we go:</p>
<p>Let $V$ be a $K$-vector space and $\dim(V)=4$. Let $B_{1}=(u_1,u_2,u_3,u_4)$ be a basis of $V$.
Let $W$ be a ... | https://math.stackexchange.com/questions/704052/about-the-matrix-of-two-linear-transformations |
Question: <p>I've been reading the <a href="https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Linear_Independence" rel="nofollow noreferrer">wikibook</a> on Linear Algebra and in the section 'Linear Independence and Subset Relations' it defines the following lemma:</p>
<blockquote>
<p>Lemma 1.14:
... | https://math.stackexchange.com/questions/1418113/linear-independence-and-subset-relations |
Question: <p>I already have the result that says that if <span class="math-container">$U$</span> is upper triangular and non singular then <span class="math-container">$U^{-1}$</span> is also upper triangular. I want to use this result to prove the result for lower triangular matrix <span class="math-container">$n \tim... | https://math.stackexchange.com/questions/1425984/if-a-lower-triangular-matrix-is-nonsingular-then-its-inverse-is-also-lower-tria |
Question: <blockquote>
<p>Let <span class="math-container">$A$</span> be a square matrix.</p>
<p>a) Show that there always exists a square matrix B such that Ker <span class="math-container">$B =$</span> Im <span class="math-container">$A$</span> and Ker<span class="math-container">$A =$</span> Im<span class="math-cont... | https://math.stackexchange.com/questions/1566424/prove-that-there-exists-a-matrix-b-s-t-kerb-ima-imb-kera |
Question: <blockquote>
<p>Let <span class="math-container">$$a = (0,2,3,-1)^T \quad b=(0,2,7,-2)^T \quad c = (0,-2,1,0)^T \quad u = (1,2,0,1)^T\quad v = (2,2,1,2)^T$$</span>
Let <span class="math-container">$U= \langle a,b,c \rangle, V = \langle u,v\rangle$</span></p>
<p>Then a) find a basis for <span class="math-conta... | https://math.stackexchange.com/questions/1398300/find-a-basis-for-u-cap-v |
Question: <blockquote>
<p>Let <span class="math-container">$T:\mathbb{R}^2\rightarrow \mathbb{R}^2$</span> such that <span class="math-container">$T(x,y)=(2x-3y,\alpha x+\beta y)$</span> and <span class="math-container">$Ker(T)=Im(T)$</span></p>
<p>find <span class="math-container">$\alpha,\beta$</span></p>
</blockquo... | https://math.stackexchange.com/questions/1429613/linear-transformation-that-imt-kert |
Question: <p>Suppose <span class="math-container">$A$</span> is similar to <span class="math-container">$B$</span> (That is: there is some nonsingular <span class="math-container">$C$</span> such that <span class="math-container">$B = C^{-1} A C $</span>). If <span class="math-container">$A$</span> is nonsingular, show... | https://math.stackexchange.com/questions/1491948/if-a-is-similar-to-b-then-a-1-is-similar-to-b-1 |
Question: <p>Suppose <span class="math-container">$A \in \mathbb{C}^{n \times n} $</span> is skew hermitian: <span class="math-container">$A^* = -A$</span>. Suppose <span class="math-container">$B$</span> is unitarily similar to <span class="math-container">$A$</span>: That is there is some unitary matrix <span class="... | https://math.stackexchange.com/questions/1496710/proving-that-a-matrix-is-skew-hermitian |
Question: <p>How do i find the a basis and dimension for $A[x]$?</p>
<p>Consider the subset of $R[x]$ given by $A[x]:=\{q(x)$ element of $\mathbb R_4[x]$ such that $q(2)=0=q(-3)\}$</p>
<p>I'm a bit confused because there are two conditions to be satisifed, $q(2)=0=q(-3)$</p>
Answer: <p>I would do the following: one ... | https://math.stackexchange.com/questions/1459238/basis-and-dimensions |
Question: <p>I'm reading the following piece of text:</p>
<blockquote>
<p>Let <span class="math-container">$T: V \to W$</span> and <span class="math-container">$S: U \to V$</span> be two linear transformations between vector spaces <span class="math-container">$U, V, W$</span> of finite dimension.</p>
<p>Since <span cl... | https://math.stackexchange.com/questions/1522119/explanation-of-notation-linear-algebra |
Question: <p>Let <span class="math-container">$A$</span> be <span class="math-container">$n $</span> by <span class="math-container">$n$</span> matrix and say <span class="math-container">$A = LU $</span> is the LU factorization of <span class="math-container">$A$</span>. Suppose <span class="math-container">$|l_{ij}| ... | https://math.stackexchange.com/questions/1469821/trying-to-establish-a-norm-inequality |
Question: <blockquote>
<p><span class="math-container">$\mathcal E_i$</span> denotes the standard basis.</p>
<p><span class="math-container">$[x]_B$</span> denotes the the coordinate vector with respect to the basis <span class="math-container">$B$</span>.</p>
</blockquote>
<p><span class="math-container">$a(1, 0) + b(... | https://math.stackexchange.com/questions/1610907/consider-the-basis-b-1-2-3-4-suppose-x-b-7-11-for-some |
Question: <blockquote>
<p>There's linear operator <span class="math-container">$A: \mathbb{R}_2[x] \to \mathbb{R}_2[x]$</span> defined as <span class="math-container">$(A(p))(x):=p'(x+1)$</span>.</p>
<p>Find all possible values for <span class="math-container">$a, b, c \in \mathbb{R}$</span> for which matrix <span clas... | https://math.stackexchange.com/questions/1828644/linear-operator-and-its-corresponding-matrix |
Question: <p>I do not understand the inversion method to solve a pair of linear equations:</p>
<blockquote>
2x<sub>1</sub> + 4x<sub>2</sub> = 4<br>
9x<sub>1</sub> + 3x<sub>2</sub> = 6
</blockquote>
<p>How to solve this? Please clarify steps.</p>
Answer: <p>Write in a matrix form $Ax=b$, i.e.
$$\pmatrix{2&4\\9&... | https://math.stackexchange.com/questions/1559659/how-to-solve-linear-equation-using-inversion-method |
Question: <blockquote>
<p>Solve the system</p>
<p><span class="math-container">$x_1 + x_2 -3x_3 = -2$</span></p>
<p><span class="math-container">$4x_1 + 3x_2 + 3x_3 = 2$</span></p>
<p><span class="math-container">$\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \begin{bmatrix}\\\\\\\end{bmatrix} + \begin{bmatrix}\\\\\\\end... | https://math.stackexchange.com/questions/1839440/solve-the-system-3 |
Question: <blockquote>
<p>Which of the following subsets of <span class="math-container">$\mathbb{R}^{3 \times 3}$</span> are subspaces of <span class="math-container">$\mathbb{R}^{3 \times 3}$</span>?</p>
<p>A. The <span class="math-container">$3 \times 3$</span> matrices with determinant 0<br/>
B. The <span class="ma... | https://math.stackexchange.com/questions/1980454/which-of-the-following-subsets-of-mathbbr3-times3-are-subspaces-of-mat |
Question: <p>I'm facing an exercise to determine basis for some spaces of polynomials. Here they are</p>
<blockquote>
<p>Consider the space of polynomials of degree equal or less than 3</p>
<p><span class="math-container">$U =$</span>{<span class="math-container">$p(t) \in \mathbb{R_3}[t]$</span> | <span class="math-co... | https://math.stackexchange.com/questions/1756963/bases-for-space-of-polynomials |
Question: <blockquote>
<p>If you add row <span class="math-container">$1$</span> of <span class="math-container">$A$</span> to row <span class="math-container">$2$</span> to get <span class="math-container">$B$</span>, how do you find <span class="math-container">${ B }^{ -1 }$</span> from <span class="math-container">... | https://math.stackexchange.com/questions/1765374/if-you-add-row-1-of-a-to-row-2-to-get-b-how-do-you-find-b-1 |
Question: <blockquote>
<p><span class="math-container">$a)$</span>Give the matrix for the Transformation <span class="math-container">$T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$</span> that first reflects points through the <span class="math-container">$x$</span> - axis and then reflects through the line <span class="ma... | https://math.stackexchange.com/questions/1840777/give-the-matrix-for-the-transformation-t |
Question: <blockquote>
<p><span class="math-container">$\forall (a,b) a,b\in \mathbb{Z}^* \exists (u,v) u,v \in \mathbb{Z}^*:a\cdot v = b\cdot u$</span> <span class="math-container">$ \land gcd(u,v) =1 $</span></p>
<p>So basically I have to prove, that for every pair <span class="math-container">$(a,b)$</span> there i... | https://math.stackexchange.com/questions/2048645/prove-that-for-every-pair-a-b-there-is-another-pair-u-v-so-that-a-cdot |
Question: <p>There are <span class="math-container">$2$</span> different bases in <span class="math-container">$R^2$</span>, <span class="math-container">$\{u_1,u_2\} , \{v_1,v_2\}$</span>. and <span class="math-container">$A$</span> is a matrix <span class="math-container">$nxn$</span>.</p>
<p>Is it possible to prove ... | https://math.stackexchange.com/questions/1720707/prove-that-av-i-bullet-av-j-v-i-bullet-v-j-forall-i-j |
Question: <p>This is a past exam question that wasn't explained in my lecture notes:</p>
<blockquote>
<p>For vector spaces <span class="math-container">$U$</span> and <span class="math-container">$V$</span> over the same field of scalars <span class="math-container">$\mathbb{F}$</span></p>
<p>Let <span class="math-cont... | https://math.stackexchange.com/questions/1777891/question-on-how-to-prove-that-a-vector-space-is-linear |
Question: <blockquote>
<p><strong>Background</strong>: Looking at properties and dealing with <a href="https://en.wikipedia.org/wiki/Matrix_(mathematics)" rel="nofollow noreferrer">Matrices</a> in linear algebra, and reading about <a href="http://mathworld.wolfram.com/MatrixInverse.html" rel="nofollow noreferrer">Matri... | https://math.stackexchange.com/questions/1861830/how-does-a-1ax-b-turn-into-x-a-1b |
Question: <blockquote>
<p>Let <span class="math-container">$M = \begin{bmatrix}8&2\\-1&5\end{bmatrix}$</span> Find formulas for the entries of <span class="math-container">$M^n$</span> where <span class="math-container">$n$</span> is a positive integer</p>
<p><span class="math-container">$M^n = ?$</span> (Shoul... | https://math.stackexchange.com/questions/1863817/finding-formulas-for-the-entries-of-a-matrix |
Question: <blockquote>
<p>Determine for which values of <span class="math-container">$a$</span> <span class="math-container">\begin{pmatrix} 4 & 0 & 0 \\ 4 & 4 & a \\ 4 & 4 & 4 \end{pmatrix}</span></p>
<p>The matrix is diagonalizable</p>
</blockquote>
<p>So we first look at the characteristic po... | https://math.stackexchange.com/questions/1902987/determine-for-with-values-of-a-the-matrix-is-diagonalizable-over-mathbbr |
Question: <p>I'm sorry that this is probably a stupid question for this page, but I have no one to ask. I'm currently studying linear algebra by myself and I'm confused by this answer:</p>
<blockquote>
<p><span class="math-container">$V$</span> is not a subspace of <span class="math-container">$\mathbb{R}_{\le4}[X]$</s... | https://math.stackexchange.com/questions/1904387/mathbbr-le3x-is-not-a-subspace-of-mathbbr-le4x-polynomials |
Question: <blockquote>
<p>Suppose <span class="math-container">$v_1, v_2, v_3$</span> are (row) vectors in <span class="math-container">$\mathbb{R}^3$</span>, and they are parallel, then what you can say about the rank of the matrix:</p>
<p><span class="math-container">\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}</s... | https://math.stackexchange.com/questions/2129639/parallel-vectors-and-rank-of-matrix |
Question: <blockquote>
<blockquote>
<p>Suppose <span class="math-container">$v_1, v_2, v_3$</span> are (row) vectors in <span class="math-container">$\mathbb{R}^3$</span>, and they are <strong>not</strong> parallel, then what you can say about the rank of the matrix:</p>
</blockquote>
<p><span class="math-container">\... | https://math.stackexchange.com/questions/2131053/rank-of-matrix-containing-non-parallel-vectors |
Question: <blockquote>
<p>Suppose <span class="math-container">$P$</span> is a plane and <span class="math-container">$x$</span> is a vector (both in <span class="math-container">$\mathbb{R^3}$</span>), can we say that</p>
<p><span class="math-container">$$x \cdot \text{proj} _{P}x = 0$$</span></p>
</blockquote>
<p>For... | https://math.stackexchange.com/questions/2131202/dot-product-of-projection-and-vector |
Question: <blockquote>
<p>Show that if <span class="math-container">$A$</span> is a diagonal matrix then orthogonal diagonalising matrix <span class="math-container">$Q = \text {Identity}.$</span></p>
<p>Proof: Let <span class="math-container">$A$</span> be a diagonal matrix and if <span class="math-container">$Q = I,$... | https://math.stackexchange.com/questions/2139992/a-question-about-a-proof-that-has-to-do-with-diagonal-matrices |
Question: <blockquote>
<p>Show that:</p>
<p>If <span class="math-container">$A$</span> is a non empty set and <span class="math-container">$R$</span> a ring, then <span class="math-container">$\operatorname{map}(A,R)$</span>, is a ring too, with the following operations:</p>
<p><span class="math-container">$f+g$</span>... | https://math.stackexchange.com/questions/2014698/structure-of-a-mapping-comes-from-the-codomain |
Question: <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be a Dedekind cut. Now let <span class="math-container">$X\oplus Y=\{x+y\mid x\in X,y\in Y\}$</span>. Show that <span class="math-container">$X\oplus Y$</span> is again a Dedekind cut. i.e. it must fulfill the following... | https://math.stackexchange.com/questions/2021755/prove-that-x-oplus-y-is-a-dedekind-cut |
Question: <blockquote>
<p>Is the following statement true or false?</p>
<p>Eliminating <span class="math-container">$z$</span> from:<br />
<span class="math-container">$x + 2y + 3z = 2$</span>,<br />
<span class="math-container">$3x + 2y + 3z = 6$</span> and,<br />
<span class="math-container">$2x + 3y = 5$</span>;<br ... | https://math.stackexchange.com/questions/2202276/true-or-false-eliminating-z-from-x-2y-3z-2-3x-2y-3z-6-and-2x-3y |
Question: <blockquote>
<p>One of the factors of <span class="math-container">$4x^2+y^2+14x-7y-4xy+12$</span> is equal to</p>
<ol>
<li><p><span class="math-container">$2x-y+4$</span></p>
</li>
<li><p><span class="math-container">$2x-y-3$</span></p>
</li>
<li><p><span class="math-container">$2x+y-4$</span></p>
</li>
<li>... | https://math.stackexchange.com/questions/2205323/to-find-factor-of-a-polynomial-equation |
Question: <p>This was a question on a recent linear algebra midterm, and I had no idea where to start.</p>
<blockquote>
<p>Fix an <span class="math-container">$m\times n$</span> matrix <span class="math-container">$A$</span> and a column vector <span class="math-container">$\mathbf{b}$</span> of size <span class="math-... | https://math.stackexchange.com/questions/2041229/how-do-i-show-that-an-equation-has-a-solution-orthogonal-to-the-nullspace |
Question: <blockquote>
<p>Theorem: Let <span class="math-container">$V$</span> be as before. If <span class="math-container">$A$</span> is an operator such that <span class="math-container">$\langle Av,v\rangle=0$</span> for all <span class="math-container">$v\in V$</span> then <span class="math-container">$A=0$</span>... | https://math.stackexchange.com/questions/2383614/hermitian-operators-langle-av-v-rangle-0-for-all-v-in-v-then-a-0-proof |
Question: <blockquote>
<p>Let <span class="math-container">$V=\mathbb{R}^{2\times 2}$</span> and define the subspaces</p>
<p><span class="math-container">$$U=\left\{\begin{pmatrix} a&0\\ 0&d \end{pmatrix}: a,d\in \mathbb{R}\right\}$$</span></p>
<p><span class="math-container">$$W=\left\{\begin{pmatrix} a&b\... | https://math.stackexchange.com/questions/2567858/direct-sum-of-two-subspace-of-mathbbr2-times-2 |
Question: <p>I'm working through the following problem.</p>
<blockquote>
<p>Let <span class="math-container">$U = \{ p \in \mathbb{P}_3 : p(1) = 0 \}$</span> and <span class="math-container">$V = \{ p \in \mathbb{P}_3 : p(-1) = 0 \}$</span>. Here, <span class="math-container">$\mathbb{P}_3$</span> represents the space ... | https://math.stackexchange.com/questions/2570880/intersection-of-two-polynomial-subspaces |
Question: <p>Let <span class="math-container">$V=\{(a,b) : (a,b) \in \mathbb{R}\}$</span></p>
<p>Is <span class="math-container">$V$</span> a vector space over <span class="math-container">$\mathbb{R}$</span> under:</p>
<p><strong>Addition</strong>: <span class="math-container">$(a_1,a_2)+(b_1,b_2)=(a_1+b_2,a_2+b_1)$<... | https://math.stackexchange.com/questions/2097193/is-v-a-vector-space-over-r-under-these-two-operations |
Question: <blockquote>
<p><span class="math-container">$P(z)$</span>, with roots <span class="math-container">$z_j$</span>'s for <span class="math-container">$0\leq j\leq a-1$</span>.</p>
<p><span class="math-container">$$P(z)=z^a+c_{a-1}z^{a-1}+\ldots+c_1z+c_0.$$</span></p>
</blockquote>
<p>I want to find the Roots of... | https://math.stackexchange.com/questions/2102294/how-to-find-the-roots-of-orthogonal-polynomial-equation |
Question: <p>This is a question from Finite-Dimensional Linear Algebra by Mark S. Gockenbach page 72 (Exercise 2.7.14). I hope to check my proof. Thank you.</p>
<blockquote>
<p>Let <span class="math-container">$V$</span> be an <span class="math-container">$n$</span>-dimensional vector space over a field <span class="ma... | https://math.stackexchange.com/questions/2585101/prove-that-the-union-of-two-bases-in-different-subspaces-is-a-basis-for-vector-s |
Question: <blockquote>
<p>A is a 3x2 matrix with two pivot positions.</p>
<p>(a) does the equation Ax=0 have a nontrivial solution</p>
</blockquote>
<p>Since the two pivot positions will create 0 in the entire column in which they are present and 1 in its own position in reduced row echelon form and the rightmost colum... | https://math.stackexchange.com/questions/2230419/nontrivial-solution-for-ax-0-and-ax-b-determine-by-pivot-positions |
Question: <blockquote>
<p>Define a rotation of <span class="math-container">$V$</span> to be a real unitary map <span class="math-container">$A$</span> of <span class="math-container">$V$</span> whose determinant is 1. Show that the matrix of <span class="math-container">$A$</span> relative to an orthogonal basis of <s... | https://math.stackexchange.com/questions/2398483/langle-av-1-av-2-rangle-ac-langle-v-1-v-1-ranglebd-langle-v-2-v-2-rangle |
Question: <blockquote>
<p>Let <span class="math-container">$V$</span> be the vector space over <span class="math-container">$\mathbb{R}$</span> of <span class="math-container">$2\times 2$</span> real symmetric matrices.
Show that the function <span class="math-container">$f$</span> on <span class="math-container">$V$</... | https://math.stackexchange.com/questions/2401585/determinant-of-a-symmetric-matrix-a-quadratic-form-proof |
Question: <blockquote>
<p><span class="math-container">$P_2(R)$</span> is the set of polynomials of degree two or lower.</p>
<p>Show that there is a unique basis <span class="math-container">$\{p_1, p_2, p_3\}$</span> of <span class="math-container">$P_2(R)$</span> with the property that <span class="math-container">$p... | https://math.stackexchange.com/questions/1949874/showing-there-is-a-unique-basis-p-1-p-2-p-3-of-p-2r-with-certain-pro |
Question: <p>I am working on a review problem for comp/qual studying and I cannot figure it out. The hint provided seems to give some intuition, but I don't see how it generalizes.</p>
<blockquote>
<p>Let <span class="math-container">$A \in GL(n,\mathbb{C})$</span> be an <span class="math-container">$n \times n$</span... | https://math.stackexchange.com/questions/1955403/eigenvalues-of-linear-operator-given-by-conjugation-by-an-invertible-matrix |
Question: <p>I wrote a proof of this (and yes, I proved the opposite direction, but I don't have a question about that portion), and I just want to get confirmation that I am not missing anything -- or advice on how to clean it up if it needs that.</p>
<p>Here's the proof:</p>
<blockquote>
<p>Suppose W is not a subset ... | https://math.stackexchange.com/questions/2114139/u-%e2%88%aaw-is-a-subspace-u-is-a-subset-of-w-or-w-is-a-subset-of-u-given-that-u-and |
Question: <p>So, I got this problem that's been bugging me. For a quick info, I'm a 12th grader in Indonesia. The problem was given by my teacher to evaluate my understanding on inequalities. Here is the problem:</p>
<blockquote>
<p>If <span class="math-container">$x$</span> and <span class="math-container">$y$</span> ... | https://math.stackexchange.com/questions/2249488/minimum-value-of-function-fx-y-if-x-and-y-are-real-numbers-and-no-other |
Question: <blockquote>
<p>For any vector space <span class="math-container">$V$</span>, subset <span class="math-container">$S \subseteq V $</span>, and vector <span class="math-container">$\vec{v} \in V$</span>, we define the set
<span class="math-container">$$\vec{v}+S = \{\vec{v} + \vec{x} : \vec{x}\in S\} $$</span>... | https://math.stackexchange.com/questions/2662417/proof-for-vectors-added-to-a-subspace-are-equal-iff-the-difference-is-in-the-sub |
Question: <blockquote>
<p>Find the characteristic polynomial,eigenvalues, and bases for the eigenspaces of the following matrices.</p>
<p><span class="math-container">$\begin{bmatrix}4&0&1\\-2&1&0\\-2&0&1\end{bmatrix}$</span></p>
</blockquote>
<p>We know that <span class="math-container">$\det(t... | https://math.stackexchange.com/questions/2415203/triangulation-of-a-matrix-and-the-eigenvalues-right |
Question: <blockquote>
<p>Let <span class="math-container">$V$</span> be a finite-dimensional vector space with (ordered) basis <span class="math-container">$\beta=(b_1,...,b_n)$</span>, and let <span class="math-container">$T:V\rightarrow V$</span> be a linear transformation. Let <span class="math-container">$B=[T]_\b... | https://math.stackexchange.com/questions/2671045/distributing-basis-coordinates |
Question: <blockquote>
<p>Are the set of vectors linearly dependent?</p>
<ol>
<li><p><span class="math-container">$ \{ e^{x}, e^{-x}\} $</span> in <span class="math-container">$\mathcal{F} (\mathbb{R} ,\mathbb{R} )$</span></p>
</li>
<li><p><span class="math-container">$ \{ \frac{1}{x-1}, \frac{1}{x + 1} \} $</span> in ... | https://math.stackexchange.com/questions/2761116/are-the-set-of-vectors-linearly-dependent |
Question: <p>I'm given two theorems:</p>
<blockquote>
<p>Theorem (1)</p>
<p>Following statements are equivalent:</p>
<p><span class="math-container">$(i)$</span> <span class="math-container">$L : V \to U$</span> where <span class="math-container">$V$</span> and <span class="math-container">$U$</span> are vector spaces,... | https://math.stackexchange.com/questions/2442512/theorems-on-1-to-1-and-onto-linear-functions |
Question: <blockquote>
<p>Let <span class="math-container">$T:\mathbb{R}^3\to W$</span> be the orthogonal projection of <span class="math-container">$\mathbb{R}^3$</span> onto the plane <span class="math-container">$W$</span> having the equation <span class="math-container">$x+y+z=0$</span>.</p>
<p>(a)Find <span class=... | https://math.stackexchange.com/questions/2326429/orthogonal-projection-of-a-point-into-xyz-0-plane-ex |
Question: <blockquote>
<p>In each one of the following cases, find <span class="math-container">$\mathscr{M}_{\beta´}^{\beta}(id)$</span>. >The vector space in each case is <span class="math-container">$\mathbb{R}^3$</span>.</p>
<p>a) <span class="math-container">$\beta=\{(1,1,0),(-1,1,1),(0,1,2)\}\\\beta´={(2,1,1),... | https://math.stackexchange.com/questions/2335624/mathscrm-beta%c2%b4-betaid-in-mathbbr3 |
Question: <blockquote>
<p>Let <span class="math-container">$C[-1,1]$</span> and <span class="math-container">$f:[-1,1]\to \mathbb{C}$</span> with the inner product <span class="math-container">$\langle f,g\rangle=\int_{-1}^{1}f(x)\overline{g(x)}dx$</span></p>
<p>Prove: <span class="math-container">$P_{0}=1, P_{1}=x, P_... | https://math.stackexchange.com/questions/2449427/checking-for-orthogonality |
Question: <blockquote>
<p>Let <span class="math-container">$\mathcal{P}_2$</span> be the space of all polynomials of degree less or equal to <span class="math-container">$2$</span> for all <span class="math-container">$f,g\in \mathcal{P}_2$</span> we define:</p>
<p><span class="math-container">$$\langle f,g \rangle=\in... | https://math.stackexchange.com/questions/2449525/showing-positive-definiteness-in-inner-product |
Question: <p>Is this map linear?</p>
<h2><span class="math-container">$T(x_1,x_2)=(x_1+2x_2+3,x_2+2x_1,3x_1)$</span></h2>
<p>Thank you very much! I thought it is not linear because there is a constant, which causes <span class="math-container">$T(v)+T(u)$</span> not to equal to <span class="math-container">$T(v+u)$</sp... | https://math.stackexchange.com/questions/2486437/is-this-map-linear |
Question: <blockquote>
<p>Find the dimension over <span class="math-container">$\mathbb{C}$</span> of the space of solutions of the following systems of equations. Also find a basis for this space of solutions.</p>
<p><span class="math-container">$ix+y-z=0\\iy+z=0$</span></p>
</blockquote>
<p>Using the formula <span cl... | https://math.stackexchange.com/questions/2361760/ixy-z-0-iyz-0-basis |
Question: <p>Let <span class="math-container">$f:\mathbb{R}^n\to\mathbb{R}$</span> be a twice continuously differentiable function such that <span class="math-container">$f(tX)=t^2f(X)$</span> for all <span class="math-container">$X\in\mathbb{R}$</span>. Show that <span class="math-container">$f$</span> is a quadratic ... | https://math.stackexchange.com/questions/2374055/derivatives-f-mathbbrn-to-mathbbr-quadratic-form |
Question: <blockquote>
<p>Let <span class="math-container">$T : \mathbb R^m \to \mathbb R^n$</span> be a linear transformation.</p>
<ol>
<li>Prove that <span class="math-container">$T$</span> is injective if and only if for every linearly independent set <span class="math-container">$\{\overrightarrow v_1,\ldots,\overr... | https://math.stackexchange.com/questions/1144193/let-t-mathbb-rm-to-mathbb-rn-be-a-linear-transformation-prove-that |
Question: <p>A bijective funktion f:V->V and a m-krylov-space K_m(f,v)=spann{v,f(v),..,f^(m-1)(v)} are given.</p>
<p>We have to show f(K_m(f,v))=K_m(f,v), so practically spann{v,f(v),..,f^(m-1)(v)}=spann{f(v),f^2(v),..,f^m(v)} if i understand it correctly.
I don't quite see how i can use the bijectivity of f here</p... | https://math.stackexchange.com/questions/3721106/show-equality-of-two-krylov-spaces |
Question: <p>Prove that exists a linear transformation <span class="math-container">$T: (Z_5)^4 \to (Z_5)^4$</span> such that:</p>
<p><span class="math-container">\begin{align}
\operatorname{Im}T &= \operatorname{Sp}\{(1,1,-1,1),(0,3,-2,2)\},\\ \operatorname{Ker}T &= \operatorname{Sp}\{(1,1,-1,1),(3,0,4,4),(3,3... | https://math.stackexchange.com/questions/3724841/prove-that-exists-a-linear-transformation-base-on-kernel-and-image |
Question: <p><strong>All the question is above <span class="math-container">$Z_7$</span>.</strong></p>
<p>I have a space:</p>
<p><span class="math-container">$$
U = \{(1,-1,1,2),(3,0,2,1)\} \subseteq (Z_7)^4
$$</span></p>
<p>I need to prove that <span class="math-container">$U$</span> is the solution space for:</p>
<p>... | https://math.stackexchange.com/questions/3725444/prove-a-space-is-a-solution-for-homogeneous-system |
Question: <p>can someone help me in seperating x in the below equation?<span class="math-container">$$\frac{x-x_l}{\sqrt{(x-x_l)^2+y_l^2}} = f*\frac{x-x_w}{\sqrt{(x-x_w)^2+y_w^2}}$$</span> <span class="math-container">$$x_l, x_w, y_l, y_w, f\ are \ constants $$</span></p>
<p>I am trying convert this into following form... | https://math.stackexchange.com/questions/3726276/solving-a-single-variable-equation |
Question: <p>My teacher made this statement in an email to me:</p>
<p>"For a complex LVS, the inner product between two vectors |u<span class="math-container">$\gt$</span> and |v<span class="math-container">$\gt$</span>,</p>
<p><span class="math-container">$\lt$</span>u|v<span class="math-container">$\gt$</span> i... | https://math.stackexchange.com/questions/3727136/does-the-inner-product-of-two-vectors-in-a-real-vector-space-have-to-be-real |
Question: <p>We multiply each entry of an <span class="math-container">$n × n $</span> matrix A by the cofactor belonging to it. What is the sum of the
<span class="math-container">$n^2$</span> terms obtained this way?</p>
<p>I dont understand how a cofactor can belong it entry. Does it mean one cofactor is same for ea... | https://math.stackexchange.com/questions/3730158/what-is-the-sum-of-the-n2-terms-obtained-this-way |
Question: <p>In the first proof at <a href="https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem" rel="nofollow noreferrer">Wikipedia</a></p>
<p>In order to write <span class="math-container">$|T(S)| = n - k$</span> in the last line of the proof, I believe we need to know each of the <span class="math-container">... | https://math.stackexchange.com/questions/3732100/question-on-proof-of-rank-nullity-theorem-on-wikipedia |
Question: <p>An <span class="math-container">$ n × n $</span> matrix <span class="math-container">$A$</span> satisfies <span class="math-container">$A^2 = 0 $</span>. Can the rank of <span class="math-container">$A$</span> be <span class="math-container">$n$</span>?</p>
<p>My opinion is that <span class="math-container... | https://math.stackexchange.com/questions/3732811/an-n-%c3%97-n-matrix-a-satisfies-a2-0-can-the-rank-of-a-be-n |
Question: <p>The the product of two matrices AB is defined if and only if the number of columns <em>n</em> in A equals the number of rows <em>m</em> in B. But what if A is a 1x1 matrix (i.e. a scalar) and B is some <em>m</em> x <em>n</em> matrix where <em>m</em> > 1?</p>
Answer: | https://math.stackexchange.com/questions/3733010/a-question-on-multiplying-matrices-by-scalars |
Question: <p>I want to prove that</p>
<p><span class="math-container">$$
\frac d{dt}\left( \det e^\mathbf {At}\right)= \text{Tr}\left(\mathbf A \right)\det e^\mathbf {At}\ ,
$$</span>
I have to use Jacobi's formula <span class="math-container">$
\frac d{dt}\left( \det \mathbf A\right)H= \text{Tr}\left( \text{adj} (\mat... | https://math.stackexchange.com/questions/3733059/help-with-jacobis-formula |
Question: <p>Construct a linear system that has no solution. Unknown Variable counts must be more than the equation count. Is it possible an equation like this? What are needed?</p>
Answer: <p>Here is an example which might be useful.</p>
<p><span class="math-container">$$
\begin{bmatrix}
1 & 1 & 1\\
2 & 2... | https://math.stackexchange.com/questions/3724488/how-to-construct-a-linear-system-that-has-no-sol |
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