Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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22 views

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length?

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length? (a) $u · u = v · v$ (b) $||u + v|| = ||u|| − ||v||$ (c) ${u \over||u||} = {v \over ||v||}$ (d) $||u + ...
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0answers
36 views

Is there a traditional name for the “eigenspace” function?

Let $A$ denote a field, $X$ denote an $A$-vector spaces, and suppose $\varphi : X \rightarrow X$ is a linear transformation. Is there a traditional name for the corresponding "eigenspace" function? By ...
2
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1answer
39 views

Matrix exponential proof

I am solving a problem: $$A^3=\alpha^2A\implies \exp(A)=E+\frac{\mathrm{sinh}\alpha}{\alpha}A+\frac{\mathrm{cosh}\alpha-1}{\alpha^2}A^2;\, \alpha\in\mathbb{C},\,A\in\mathbb{M}_{n\times ...
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3answers
61 views

How can I show that $x^TAx=\mathrm{tr}(Axx^T)$? [duplicate]

How can I show that $x^TAx=\mathrm{tr}(Axx^T)$? $A$ is an $n$-dimensional square matrix and $A^T$ is the transpose of $A$.
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1answer
34 views

Uinviersal property of basis of a vector space

Let V be a vector space over a field k. Let B be a subset of V. If any set map from B to any vector space W can be extended uniquely to a k-linear map from V to W. Then B is a basis of V. Can ...
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2answers
28 views

Why not $R_j + cR_i \rightarrow R_i$ for Elementary Row Operation of Replacement ? [Lay P6]

P6 of Linear Algebra and Its Applications, 4th Ed by David Lay says: Replacement: $\color{green}{kR_j + R_i \rightarrow R_i}$. For example (from BP P435 Example 2): For example (from BP P435 ...
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1answer
27 views

number of solutions and rank

Consider a matrix A of $a\times b$. If we know the how the rank of the matrix is related to a and b, we can determine (maybe not exactly) the number of solutions for the system. Now, if we know the ...
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3answers
42 views

How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$

How can I show that $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$ for any natural number $n$ and $x_i \in\mathbb{R}?$ I assume there is something about Cauchy Schwartz and induction, but I really ...
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3answers
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how does $\det((\det A) I)= (\det A)^n$

Taking determinants of both sides of $A (\text{adj} A) = (\det A) I$, we have $$\det((A) (\text{adj} A))= \det((\det A) I) \text{ or } (\det A) (\det(\text{adj} A)) = (\det A)^n$$ When looking at the ...
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1answer
74 views

Generalized Cross Product

I know that the cross product can be generalized as $$\text{cross}(x_0,...,x_{n-1})=\det\begin{vmatrix}&x_0&\\&x_1&\\&\vdots&\\e_1&\cdots&e_n\end{vmatrix}$$ where $e_i$ ...
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1answer
43 views

Is there a significance to a matrix having an eigenvector equal to a column vector within a matrix?

Consider matrix $A$: $\begin{bmatrix} 2 & -2\\ 3 & -3\\ \end{bmatrix}$ After little computing, we find the eigenvectors (and their corresponding eigenvalues) to be equal to $E_{\lambda=0}= ...
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1answer
25 views

Linear Algebra nullSpace and multiplication

If you have a $m \times n$ matrix $A$, and an $n \times p$ matrix $B$ and $AB=0$. Is the $dim (null A) = p$? No clue how to start this.
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0answers
16 views

Covariance matrix with constant diagonal

Is there a term for covariance matrices with constant diagonal (variance of every entry being equal)?
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2answers
35 views

Proving That a $1 \times 1$ Matrix has a Rank of $1$

I have an assignment that asks me to prove something but I've hit a roadblock. Let $u$ be a $3 \times 1$ matrix with $u_{1}$, $u_{2}$, and $u_{3}$. Let $v$ be a $3 \times 1$ matrix with $v_{1}, ...
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1answer
34 views

Distance of a point to a plane

Let $T$ be the plane $x+2y+3z=11$. Find the shortest distance $d$ from the point $P=(2, 4, 5)$ to $T$, and the point $Q$ in $T$ that is closest to $P$. This is just one of the questions on my ...
3
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4answers
186 views

If all eigenvalues are 1 or -1, is then $A^{12}=I$?

True or false: If all the eigenvalues of A are either $\lambda=1$ or $\lambda = -1$ then $A^{12}$= I If we have a matrix $$\mathbf A = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ this has ...
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1answer
16 views

For all non-zero vectors v in R^n, the non-zero vector u is orthogonal to what?

I have a multiple choice question: ...
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0answers
12 views

Books in spectral theory for finite dimensional spaces

I'm looking for beginner books of spectral theory for finite dimensional spaces. I've already heard about this subject, but I don't know where I can find it. What's the domain of this subject? (Linear ...
1
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1answer
27 views

Prove trace inequality $\mathrm{tr}\{ABCBAD-ABCD-ADCB+CD\} \geq 0$

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be four (generally non-commuting) positive semidefinite matrices of same size. I want to show that (or find a counterexample to) $$ ...
4
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4answers
196 views

Every n × n-matrix A with real entries has at least one real eigenvalue. [duplicate]

I have a true/false question: Every n × n-matrix A with real entries has at least one real eigenvalue. I am thinking that this is true but I would like to hear ...
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0answers
13 views

Find a basis and a formula for the coordinate map.

Let $W = \{f(x) \in P_{2} | f(2) = 0\}$ is a subspace of $P_{2}$. Find a basis and formula for the coordinate map from $W to \mathbb{R}^{2}$. I think I found the basis to be $\{4,2,1\}$, I don't ...
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1answer
19 views

If v1, v2 are two non-zero vectors in R^3, span{v1, v2} is a plane through the origin

I have a true/false question If v1, v2 are two non-zero vectors in R^3, span{v1, v2} is a plane through the origin This is true since it falls under the ...
2
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2answers
47 views

Is it possible to triangularize a matrix only by adding scalar multiples of rows to each other?

I am working on showing if $B$ is a $s \times s$ matrix, $D$ is a $t \times t$ matrix, $C$ is a $s \times t$ matrix, and $0$ is a $t \times s$ zero matrix, then $\det(A)=\det(B)\det(D)$, where $$A = ...
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2answers
62 views

Is $W$ a subspace of $\mathbb{R}^{n \times n}$?

Let $W = \{A \in \mathbb{R}^{n \times n} | A_{11} \geq 0\}$ is $W$ a subspace of $\mathbb{R}^{n \times n}$? Prove or disprove. I know how to do it if it was a specific sized matrix, but I'm not sure ...
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2answers
34 views

Show that T is a linear transformation.

Let B be an element of $R^{n \times n}$ and define $T(A) = BAB$ for all $A \in R^{n \times n}$. Show that T is a linear transformation. I am completely lost and I do not know how to start this.
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1answer
24 views

A question about the minimal polynomial of a transformation.

My Linear Algebra textbook states: Say the minimal polynomial of the transformation $T$ is $m(x)=p_1(x)^{e_1}p_2(x)^{e_2}\cdots p_s(x)^{e_s}$. Then $$n(p_i(T))^{e_1}$$, which is the null space ...
2
votes
1answer
36 views

Max determinant

Working on a 5x5 matrix the max determinant I figured out so far is 4.The entries I'm using is 0 and 1 . Is that the only determinant or is there bigger determinant than 4?
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1answer
24 views

vector projection on to subspaces

If given a subspace of R4 that is spanned by the set of orthogonal vectors W =span { (0,1,1,1),(1,1,0,-1) }. How to find the projection of a vector u onto the subspace? if u = (2,1,2,0)? What I have ...
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1answer
27 views

n by n matrices in $\mathbb C$ are either diagonalizable or similar to following matrix.

I have just proven that a Matrix $A \in M_{2}(\mathbb C)$ is either diagonalizable or similar to a matrix $B=\begin {pmatrix}\lambda &1 \\0& \lambda \end {pmatrix}$ $\lambda \in \mathbb C$. I ...
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1answer
55 views

How to show that T is diagonalizable iff $\dim S_{\lambda_1}+\dim S_{\lambda_2}+\dots+\dim S_{\lambda_k}=\dim V $

Theorem: $V$ is a vector space on field $F$. and $T:V\to V $ is linear transformation. $\lambda_1,\lambda_2,\dots,\lambda_k$ are eigenvalues and $ S_{\lambda_1},S_{\lambda_2},\dots,S_{\lambda_k} ...
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3answers
55 views

Function composition $f(g(x)) = x$

Let $f: N \rightarrow N_0$, where $f(x) = x-1$; and $g: N \rightarrow N_1$, where $g(x) = x+1$; $$(f \circ g)(x) = x?$$ Can anyone explain why/if this is true? Shouldn't it be $x^2$?
4
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2answers
78 views

How to show that there is no $3\times3$ real matrix $A$ such that $A^2+I=0$?

Question: show that there is no $3\times3$ real matrix $A$ such that $A^2+I=0$? Is it because: $$\det(A^2)=\det(-I)\\ \implies \det(A)\det(A)=-1\\ \implies \det(A)=-i$$ How to continue?
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1answer
18 views

When is a symmetric 2-tensor field globally diagonalizable?

Suppose that $\mathbb{R}^n$ has a Riemannian metric $g$. Let $h$ be a smooth symmetric 2-tensor field on $\mathbb{R}^n$. At any point $p \in \mathbb{R}^n$, there is a basis of $T_p \mathbb{R}^n$ in ...
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1answer
29 views

Prove that $\chi_{V_1 \otimes V_2} (g) = \chi_{V_1} (g) \cdot \chi_{V_2} (g).$

Here, $\chi$ is the character of the sub-representation, i.e., Given $\rho : G \to GL(V)$ is a representation, then the function $\chi_{\rho}: G \to \mathbb{C}: \chi_{\rho}(g) \to Tr(\rho_g)$. I ...
5
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3answers
36 views

Does the rank-nullity theorem hold for infinite dimensional $V$?

The rank nullity theorem states that for vector spaces $V$ and $W$ with $V$ finite dimensional, and $T: V \to W$ a linear map, $$\dim V = \dim \ker T + \dim im T$$ Does this hold for infinite ...
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1answer
33 views

$\det(A) = \det(A^T)$ for elementary matrix.

We proofed in class that for any matrix $\det(A) = \det(A^T)$. I was asked to prove the same, only for elementary matrices. Though repeating the proof for any matrix would do the work, it's like using ...
2
votes
1answer
33 views

Find linear transformation given kernel

$ F: R^4 -> R^3 $ $ kerF=span\{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}, \begin{bmatrix}0\\1\\1\\1\end{bmatrix} \} $ Find linear transformation in canonical bases given above information. I tried ...
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3answers
52 views

Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
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1answer
11 views

what is the dimension of this subspace for given problem

In a subspace $W=\{[a_{ij}]:a_{ij}=0$ if $i$ is even$\}$ of all $10\times 10$ real matrix, what is the dimension of W?
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1answer
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How can i show that $x^TAx=\mathrm{tr}(xAx^T)$? [closed]

How can i show that $x^TAx=\mathrm{tr}(xAx^T)$? A is square matrix. A^T is transpose of A.
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2answers
35 views

Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A^T}) = \sum\limits_{\pi \in {S_n}} ...
2
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3answers
43 views

Evaluate determinant of an $n \times n$-Matrix

I have the following task: Let $K$ be a field, $n \in \mathbb{N}$ and $a,b \in K^n$. Evaluate the determinant of the following matrix: $$\begin{pmatrix} a_1+b_1 & b_2 & b_3 & \dots ...
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4answers
104 views

Multiplying Adjacent Matrices?

My teacher hasn't explained it too well, so i'm looking for an explanation: $$A = \begin{pmatrix} 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0 \end{pmatrix}$$ $$A^2 = ...
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1answer
20 views

Showing an endomorphism is not surjective

Let $$A=\begin{pmatrix}2&-2\\2&-2\end{pmatrix}$$ and the endomorphism $f_A:M_2(\mathbb R)\longrightarrow M_2(\mathbb R); B\longmapsto AB$. I want to show that $f_A$ is not surjective. My try: ...
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2answers
37 views

Is $I+AA^T$ positive definite matrix?

If $A$ is real matrix, how can i show that $I+AA^T$ is positive definite matrix? $I$ is the identity matrix and $A^T$ is a transpose of $A$.
2
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1answer
26 views

Find conditioning of the matrix

Find conditioning of the following matrix: $$A=\begin{bmatrix}1& 0\\1&\epsilon\end{bmatrix}.$$ in a $\|.\|_\infty$ norm for $\epsilon > 0$
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1answer
27 views

System of equations with insufficient equations [closed]

I have a system of equations problem with $5$ variables but only $4$ equations: Suppose that $x,y,z,u$ and $v$ are real numbers that satisfy the following system: $$ \begin{align} -4x + 6y + 14z ...
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3answers
44 views

Problem with LU decomposition

I have this matrix: $$ A =\begin{bmatrix}1 & -2 & 3\\ 2 & -4 & 5 \\ 1 & 1 & 2\end{bmatrix} $$ After I decomposit it, I get: $$ L = \begin{bmatrix}1 & 0 & 0\\1 & 1 ...
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0answers
20 views

Fixed points and permutations.

Let $\psi ,\varphi \in {S_n}$ two permutations. Let $M$ a matrix such that $a_{i,j}=1$ iff $i=\sigma(j)$ where $\sigma \in S_n$ ($0$, otherwise) I already showed that $tr(M) = \left| {\left\{ {k \in ...