Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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How can we turn $V = \begin{bmatrix} 0 + x_2 + x_3\\x_1 + 0 + x_3 \\x_1 + x_2 + 0 \end{bmatrix}$ into a vector multiply by a matrix?

Given $V = \begin{bmatrix} 0 + x_2 + x_3\\x_1 + 0 + x_3 \\x_1 + x_2 + 0 \end{bmatrix}$ What is a way to decompose this vector into a vector multiplying a matrix? i.e. $V = M x$, where $ x = [x_1, x_2,...
2
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1answer
16 views

How to represent matrix multiplication in tensor algebra?

How can we represent matrix multiplication in tensor algebra? Even if we assume all matrices represent contravariant tensors only, clearly matrix multiplication does not correspond to the ...
4
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1answer
25 views

Significance of symmetric characteristic polynomials?

By symmetric characteristic polynomial, I mean for example... the characteristic polynomial of the $3\times3$ identity matrix is: $x^3 - 3x^2 + 3x - 1$ similarly for the $4\times4$ identity matrix ...
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1answer
14 views

Finding linear functionals on $\mathbb R^4$ the intersection of whose null spaces is a linear span.

Find two linear functionals in $\mathbb R^4$ the intersection of whose null spaces is the linear span of $(1,1,1,1)$ and $(1,0,-1,0)$. You now have in hand a linear transformation whose null space is ...
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1answer
24 views

Let A and B be n by n matrices . Prove that if A is symmetric and B be skew-symmetric , then {A,B} is a linearly independent set.

can anybody help me plz? Let $A$ and $B$ be $n \times n$ matrices ($A$ and $B$ are not $0$) . Prove that if $A$ is symmetric and $B$ be skew-symmetric , then $\{A,B\}$ is a linearly independent set.
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2answers
27 views

Find $\lambda$ and $\theta$ such that it validates this matricial equation

Find $\lambda$ and $\theta$ such that it validates the matricial equatial $$ \left( \begin{array}{cc} 1 & 2 \\ 2 & 3 \end{array} \right) % \left( \begin{array}{cc} \cos \theta \\ \sin \theta ...
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2answers
40 views

How to find largest coefficient in matrix?

$$\begin{bmatrix} 1&2&6\\ 7&8&3\\ 0&4&7 \end{bmatrix} $$ I want know the algorithm to find largest value in matrix .
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1answer
40 views

Can someone offer a way to simplify $x_1(y_1 - x^Ty) + x_1(w_1 - x^Tw)^2 - x_1^2(w_1 - x^Tw)^2 + x_1x_2(w_1 - x^Tw)^2$

Let $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, $w =\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ I have the following vector: $V = \begin{bmatrix} ...
1
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1answer
16 views

Proportionality between two quantities

Its known that if one variable is proportional to two others than it is also proportional to their product. $$\forall a,b,c\in ℝ:a\propto b\wedge a\propto c\Rightarrow a\propto b\cdot c$$ I think i`ve ...
6
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3answers
151 views

Are there ways to solve equations with multiple variables?

I am not at a high level in math, so I have a simple question a simple Google search cannot answer, and the other Stack Exchange questions does not either. I thought about this question after reading ...
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0answers
29 views

Why does the gradient of matrix product $AB$ w.r.t. $A$ equal $B^T$?

The below passage is from p. 215 of Deep Learning by Goodfellow, Bengio and Courville. For example, we might use a matrix multiplication operation to create a variable $C = AB$. Suppose that the ...
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2answers
34 views

show isomorphism by using characteristic polynomial

Let $A\in M_2\left(\mathbb{C}\right)$ be with two different eigenvalues $t_1,t_2$ I have got to show the following isomorphism: $\;\left.\mathbb{C}\left[x\right]\middle/ p_A\left(t\right)\right.\...
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4answers
54 views

What does this sentence mean? “$\lim_{x\to x_0}$ exists at every point $x_0$ in (-1,1)$.”

What does this sentence mean? $$\lim_{x\to x_0} \;\text{exists at every point}\; x_0 \; \text{in} \; (-1,1).$$ $(1,-1)$ is just an example point. The topic is finding whether limit functions ...
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1answer
23 views

rank of a vector space [on hold]

I read this sentence in a report concerning in symmetric cone programs: "Let J be a Euclidean Jordan algebra with dimension n, and rank r." I know what the rank of a (matrix) is.. does the rank here ...
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0answers
20 views

numerical recipes in c , nrutil.c , the matrix function [on hold]

Why the matrix function contains : m += NR_END; m -= nrl; What exactly does these two lines do ? Also why , m is allocated with (nrow+NR_END) , why not just nrow ?
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0answers
23 views

Equivalence between trace and Euclidean norm

In the paper "On best approximate solutions of linear matrix equations", there is a very small equivalence I don't know where it comes from. Let $A$ be a matrix (either real or complex), and $\|A\|$ ...
1
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1answer
50 views

How to solve for P in a similarity when matrices aren't diagonalizable

When matrices $A,B$ are similar and they are both diagonalizable, as $$ \left\{ \begin{array}{l} P_1^{-1}AP_1=Λ\\ P_2^{-1}BP_2=Λ \end{array} \right. \Longrightarrow (P_1P_2^{-1})^{-1}A(P_1P_2^{-1})=B$$...
1
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1answer
21 views

Summation functional on a Hamel basis

Let $X$ be an infinite-dimensional Banach space. Is it possible to choose a Hamel basis $B$ of $X$ such that the linear functional defined by $f(b)=1$ ($b\in B$) was continuous?
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1answer
30 views

find minimum of a matrice

Let $f~:~\Bbb R^2\to \Bbb R$ be defined as: $$f(x)=\left\|\begin{bmatrix}2&1\\3&1\\4&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix} - \begin{bmatrix}2\\1\\7\end{bmatrix}\right\|_2^2$$ ...
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1answer
43 views

Is a substitution of variable a linear transformation?

I've been asking myself this. A linear transformation $T:U \rightarrow V$ between the vector spaces $U$ and $V$ is a function that respects the following: a) $T(x+y) = T(x)+T(y) $ $ \forall x,y \in ...
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1answer
49 views

$A$ unipotent and $A^k(U)=U$ for a subspace $U\subset \mathbb{C}^n$, does $A(U)=U$?

Let $A$ be a unipotent $n\times n$ complex matrix. Let $k\geq 1$ be an integer. Let $U$ be a subspace of $\mathbb{C}^n$ such that $A^k(U)=U$. Does this mean that $A(U)=U$? EDIT: A matrix $M$ is ...
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1answer
28 views

Is change of basis a linear operation?

If $T$ and $U$ are linear operators on $V$ and $\beta$ is a basis for $V$, is it true that $$[T + aU]_\beta = [T]_\beta + a[U]_\beta\ ?$$ I know that scalar multiplication is preserved, just not sure ...
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0answers
15 views

Unit ball in normed vector spaces [on hold]

Let $\mathrm{B}$ be a unit ball of some norm in vector space $\mathrm{L}$, $\mathrm{B^*}$ - a unit ball of induced norm in $\mathrm{L^*}$ = $Hom( \mathrm{L}, \mathfrak{K}), \mathfrak{K} = \mathbb{R}~ ...
1
vote
1answer
30 views

lower bound for the trace$(A^\dagger A).$

Let $A$ be $n\times n$ complex squared matrix. I want to find a lower bound for $\mathrm{tr}(A^\dagger A).$ What I could find so far is that if $A$ is Hermitian then $$\mathrm{tr}(A^\dagger A) \geq \...
0
votes
1answer
17 views

How to find rank of any matrix?

I'm asking this as a general question. Usually when I'm asked to find rank I am totally confused as to how to proceed at all. None of the transformations behind don't make sense to me. Is there a ...
1
vote
1answer
48 views

Is it true that $SL(n, \mathbb R)=<\{ABA^{-1}B^{-1} : A,B \in GL(n,\mathbb R) \}$ >? [on hold]

Is it true that $\operatorname{SL}(n, \mathbb R)=\left\langle \left\{ABA^{-1}B^{-1} : A,B \in \operatorname{GL}(n,\mathbb R) \right\} \right\rangle$?
2
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1answer
27 views

Vector spaces as bimodules

The usual definition of a vector space $V$ over $K$ is as an abelian group, on which $(K\setminus\{0\},\cdot)$ acts on the left, such that furthermore the operation of $K$ on $V$ is compatible with ...
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0answers
5 views

Characterization of the feasible set of an optimization problem

Let $V \in \mathbb{R}^{n\times n}$ be a positive definite matrix and $D \in \mathbb{R}^{k\times k}$ a diagonal matrix with strictly positive elements on its diagonal. We also have a matrix $X \in\...
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0answers
30 views

Determinant of a Matrix and Determinant of Its Transpose Question [on hold]

Why does the determinant of a transposed matrix have a matching inverse cycle for each permutation of the non-transposed matrix?
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1answer
21 views

The dimension of $:W_1\cap W_2$

$$W_1=\operatorname{span}\left(\begin{pmatrix}1&1\\ 0&0\end{pmatrix},\begin{pmatrix}3&1\\ -1&0\end{pmatrix}\right)$$ $$W_2=\operatorname{span}\left(\begin{pmatrix}1&1\\ 1&0\...
0
votes
2answers
40 views

Find $\lambda $ so the dimension of the vector subspace is 2.

$\left\{a,b,c\right\}\in \mathbb{R}^3$ are linearly independent vectors. Find the value of $\lambda $, so the dimension of the subspace generated by the vectors: $2a-3b,\:\:\left(\lambda -1\right)b-...
1
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3answers
46 views

Solve the system for the given parameter a

\begin{align} ax+y+z&=1,\\ 2x+2ay+2z&=3\\ x+y+az&=1 \end{align} I tried forming the system matrix and discuss it using its rank, but I'm not sure how to row reduce: $$\begin{pmatrix}a&...
1
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1answer
22 views

Prove the invertibility of $X^T X$ when $X$ is a (rectangular) Toeplitz-like matrix.

In order to use a minimum squares estimator over some discrete dynamic system parameters, it is necessary to prove that the product $X^T X$ is invertible. Consider the following $N$ by $n+1$ matrix $X$...
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2answers
48 views

Are the matrices A and B similar?

$B=\begin{pmatrix}1&7&0\\ \:0&2&7\\ \:0&0&2\end{pmatrix}$ $A=\begin{pmatrix}1&1&5\\ 0&2&0\\ 0&0&2\end{pmatrix}$ They have the same trace,same rank, ...
0
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3answers
37 views

Show that similar matrices have same trace

If $A$ and $B$ are $n\times n$ matrices of a field $F$, then show that $\text{trace}(AB)=\text{trace}(BA)$. Hence show that similar matrices have the same trace. I've done the first part (proving ...
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1answer
22 views

Show that $|\lambda_i(A)|<1$ iff $|\lambda_i(\beta A)|<1$ $\forall \beta: |\beta|\leq 1$

Here $\lambda_i(A)$ is the $i$-th eigenvalue of the square matrix $A$. I would like to know if these two inequalities are equivalent. I assumed they are (please correct me if I am wrong). So I tried ...
4
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6answers
120 views

Prove that $A$ cannot be invertible if $A^2=0$

Let $A$ be an $n\times n$ matrix for which $A^2=0$. Prove that $A$ can not be invertible. My attempt: Given $A^2 = 0$, this means that $A = 0$. If $A$ is invertible, there must be an $n \times n$ ...
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0answers
21 views

Flip only one axis of a transformation matrix?

A 4x4 transformation matrix can be multiply to transform a point by translating and rotating it. I have a transformation matrix, however I noticed that the X translation is going to the opposite way ...
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2answers
33 views

Finding the expression of the inverse of $(AB)^T$

I know that $(AB)^T$ = $B^TA^T$ and that $(A^T)^{-1}= (A^{-1})^T$ but couldn't reach any convincing answer. Can someone demonstrate the expression.
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3answers
26 views

Solving the matrix equation and finding the matrices inverse

Suppose that $E_1 \begin{bmatrix}12\\35\end{bmatrix} = \begin{bmatrix}48\\35\end{bmatrix}$ Find $E_1$ and $E_1^{-1}$ I set up the equation and saw that $E_1 * A = B$ so I know that: $E_1 = A^{-1} * ...
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3answers
31 views

Finding possible determinant values of 3x3 matrix using an equation

Given a 3 x 3 matrix $A$ $4A= A^{7}$ Find the possible values of det(A). I multiplied by $A^{-1}$ both sides and got $4I= A^{6}$ (not helpful) ?? Can you show the right way to solve it ? and how ...
2
votes
1answer
60 views

“Universal property” of cross product

Let $V$ be a three dimensional euclidian vector space which is oriented. Because of the orientation, we can define the cross product $\times: V^2 \rightarrow V$ uniquely by: $<v\times w,u> = \...
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1answer
34 views

Show that $\widetilde{f}$ is a nondegenerate map?

Noted by $V, W$ two vectors spaces over the same field $K$ of finite dimensions. Let $f:V\times V\rightarrow W$ a degenerate map. I would show that $$\widetilde{f}:\widetilde{V}\times \widetilde{V}\...
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1answer
37 views

Relationship between averages of $x^tx$ and $xx^t$ for column vector $x$

If we have data set $x$ as $m$ of $n \times 1$ vectors, and we know the average over index $m$ of $xx^t$ is $<xx^t> = C$, where $C$ is $ n \times n$ matrix. What is the average of scalar $ x^...
2
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1answer
28 views

Linear four-parameter recurrence from Concrete Mathematics

In the book Concrete Mathematics, there's an exercise (1.16) where you're asked to solve a general four-parameter recurrence using the Repertoire Method. The recurrence is defined as follows: \begin{...
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0answers
9 views

Sample points from a multivariate normal distribution using only the precision matrix?

I have a problem where I can directly compute the (sparse) precision matrix (inverse of the covariance) of a multivariate normal distribution, but the covariance itself is not sparse and I don't want ...
1
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3answers
45 views

Linear operator proof.

Let $P$ and $Q$, respectively, be subspaces of the vector spaces $V$ and $W$ over the same field $K$ and let $V$ be a finite dimensional space. If $\dim P + \dim Q = \dim V$ prove that then there ...
0
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1answer
24 views

Let $A^*$ denote the matrix whose $(ij)$-th entry is $A_{ij}$, $1 ≤ i, j ≤ 5.$

Let $A \in M_5(\Bbb R)$. If $A = (a_{ij})$, let $A_{ij}$ denote the co-factor of the entry $a_{ij}, 1 ≤ i, j ≤ 5.$ Let $A^*$ denote the matrix whose $(ij)$-th entry is $A_{ij}$, $1 ≤ i, j ≤ 5.$ a. ...
2
votes
1answer
36 views

Which is true about $Q$ where $Q=I+2P$

Let ${a_{1},a_{2},...a_{n}}$ and ${b_{1},b_{2},...b_{n}}$ be two bases of $\mathbb{R}^{n}.$ Let P be an $n \times n$ matrix with real entries such that $Pa_{i}=b_{i}$ for $i=1,2, ...,n.$ Suppose that ...
0
votes
1answer
17 views

difference between linear map basis and vector basis

A linear map can be represented as a matrix in a certain basis P. Similarly, given a vector space over a field, its basis can be found, say Q. How is the concept of P related to that of Q? Are they ...