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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Theorems restricting the eigenvalue of a matrix

I have a square matrix $C$, whose entries I will denote by $c_{ij}$. Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows: $$ c_{ij} = s_{ij}/S_i $$ The $s_{ij}$ and $S_i$ satisfy the ...
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How to prove that $W_1\cap W_2\supset$ Span$(S_1\cap S_2)$ if $W_1=$ Span$(S_1)$ and $W_2=$ Span$(S_2)$ are subspaces of vector space?

In my opinion, let $v\in$ Span($S_1\cap S_2$) and therefore $v\in$ Span$(S_1)$ and $v\in$ Span$(S_2)$. Write $v=c_1z_1+...+ c_nz_n$ where $z_k\in S_1\cap S_2$ and $c_k\in R$. Here I am feeling I have ...
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1answer
16 views

Almost completing the argument: $p: L \to L$ a projector, then $im~ p \oplus \ker p = L$.

I had to show that if $p: L \to L$ is a projector, then $im ~p \oplus \ker p = L$. This was easy. Now I have to show that the matrix of $p$ is divided on four blocks where one of them is $r$ ...
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1answer
26 views

If commutativity of vector space is omitted, can we still use other axioms to prove the commutativity?

Here I am thinking of using $-(x+y)$ and show that it equals $-(y+x)$. $-(x+y)=-x-y$ by distributivity =$-x-y+0=...$ Here I don't know how to continue, could someone suggest?
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Derivative with respect to vectors related through a matrix

Consider a function $g: \mathbb{R}^r \to \mathbb{R} $ and two vectors $\mathbf{b} \in \mathbb{R}^r$ and $\mathbf{c} \in \mathbb{R}^m$ such that $\mathbf{c} = \mathbf{A}\mathbf{b}$. If I calculate the ...
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1answer
16 views

How to show that the following is satisfied for all vector space axiom?

Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$ ...
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17 views

Check if two square matrices are similar.

Check if matrices $A= \begin{bmatrix} 1 & 1 & 5 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 7 ...
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Can we attach a space with discrete signal?

This question refers to the link https://en.wikipedia.org/wiki/Space_(mathematics) and https://en.wikipedia.org/wiki/Discrete-time_signal. My question is how can we associate a discrete signal with a ...
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1answer
15 views

Showing that $O$ is the only nilpotent matrix in $\langle A \rangle$ where $A$ is diagonalizable

I have the following task: Let $A\in \mathcal{M}_n(K)$ be a diagonalizable square matrix. Show using the spectral decomposition of $A$ that the only nilpotent matrix in $\langle A\rangle ...
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17 views

is there a closed form expression for the following matrix infinite series

Consider this infinite sum of matrices. Is there any closed form to express this sum? $S=B+ABA^T +A^2B({A^T})^2+A^3B({A^T})^3+...$ And B is diagonal. Thanks
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1answer
21 views

Prove that $u$ is orthogonal to $v-\operatorname{proj}u(v)$ for all vectors $u$ and $v$ in $\mathbb{R}^{n}$ where $u \neq 0$.

Here's where I'm at, not sure where to go from here. Two vectors are orthogonal if their dot product is $0$. Knowing that; $$u \cdot (v - \operatorname{proj} u(v)) = 0$$ $$u \cdot \left(v - \frac{u ...
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8 views

Using operators to test for eigenstates of Gaussian wavepacket?

Let's say I have some state $|\alpha\rangle$ that is represented in the position basis of a Gaussian wavefunction. Following Sakurai's notation, where $N$ is the normalization constant we have ...
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1answer
22 views

How to show that $M_{2\times 2}(\mathbb{R})=W_1\oplus W_2$ based on the following assumption?

Let the subspaces $W_1=\{\begin{pmatrix}a&b\\-b&a \end{pmatrix}|a, b\in \mathbb{R}\}$ and $W_2=\{\begin{pmatrix}c&d\\d&-c \end{pmatrix}|c, d\in \mathbb{R}\}$ of $M_{2\times ...
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1answer
9 views

If $x+y=(x_1y_1, …, x_ny_n)$ and $c\cdot '\ x=x^c_1, …, x^c_n$, how to show that with these two operation $V$ is a subspace?

Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., ...
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19 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...
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An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
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1answer
18 views

Inner product in a direct sum of a dimensional space

Supposer that $V = W_{1} \oplus W_{2}$, $f_{1}$ and $f_{2}$ are inner product at $W_{1}$ and $W_{2}$, respectively. Show that there is only one inner product $f$ in $V$ such that i) $W_{2} = ...
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2answers
36 views

showing projection is a linear operator

Show that the orthogonal projection is linear. Let $x_i=y_i+z_i$, where $x_i\in X$, $y_i\in Y$, $z_i\in Y^\perp$, and $\alpha,\beta$ be scalars. Then \begin{align}P(\alpha x_1+\beta ...
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Transformation matrix from principal angles and vectors

If I got it right, given two planes in $N$-dimensional space ($N\gg2$), their 2 principal angles ($\theta_1$, $\theta_2$) and 4 vectors ($\vec{a}_1$, $\vec{a}_2$, $\vec{b}_1$, $\vec{b}_2$) can be ...
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Orthogonal projections exercise

Let $V$ be a $n-$dimensional space with inner product and consider $W$ a subspace of $V$. If $E$ it's a projections with $Im E = W$ such that $|E\alpha| \leq |\alpha|$ $\forall \alpha \in V$ then $E$ ...
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1answer
26 views

Basis for 4th degree polynomials such that integral of $p(x)$ from $-1$ to $1$ equals $0$

Let $U= \{ p \in \mathscr P_4\mathbb{R} \ | \int_{-1}^1 p(x)dx=0\}$. a.) Find a basis for $U$. b.) Find a subspace $W$ of $\mathscr{P_4}(\mathbb{R})$ such that $\mathscr{P_4}(\mathbb{R})= U \oplus ...
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24 views

In matrix algebra, what's the name for the inverse operation of pre- or post- multiplication?

For example, in this typical equation: $$\mathbf{Mv}-\lambda \mathbf{v}=\mathbf{0}$$ (where $\mathbf{M}$ is a symmetric matrix, $\mathbf{v}$ is a vector, $\lambda$ is a scalar, and $\mathbf{0}$ is a ...
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2answers
20 views

Proving that $u$ and $v$ are linearly independent, given the independence of $T(u)$ and $T(v)$

Suppose that $T$ is a linear transformation and that $T(u)$ and $T(v)$ are linearly independent. Prove that $u$ and $v$ are linearly independent. I have no idea where to start in this case. Just need ...
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1answer
10 views

Finding all eigenvectors and eigenvalues of a linear operation on a function

Here is the question I am stuck on: Consider $T \in \mathrm{Hom}(\Bbb{R}[x]_{\le 2} ,\Bbb{R}[x]_{\le 2} )$ given by $$ (Tf)(x)=\int_{-1}^1(x-y)^2f(y)dy-2f(0)x^2$$ for all $f \in \Bbb{R}[x]_{\le ...
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1answer
33 views

linear algebra in infinite dimension

I look for an advanced linear algebra (A complete book but wich deals indiferently with infinite/finite vector space). To give an idea i expect a book that (for exemple) would prove the existence of a ...
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1answer
23 views

For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?

Is it true that for every integer $n>1$ , there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $A \in M(n,\mathbb R)$ and $AD=DA \implies A$ is also a diagonal matrix ?
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1answer
14 views

Let $n>1$ and $g_1,…,g_{n-1}$ be $C^2$ scalar fields over $\mathbb R^n$ , then for any scalar field $f$ , is $\det J(f,g_1,…,g_{n-1})=0$?

Let $n>1$ and $g_i:\mathbb R^n \to \mathbb R$ be scalar field for each $1\le i\le n-1$ such that all second order partial derivatives of each $g_i$ exist and are continuous ( i.e. each $g_i$ is ...
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23 views

Derivative of $l_1$ norm

I want to compute the following derivative with respect to $n\times1$ vector $\mathbf x$. $$g = \left\lVert \mathbf x - A \mathbf x \right\rVert_1 $$ My work: $$g = \left\lVert \mathbf x - A ...
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20 views

approximate projection into eigenvector space

Given a matrix A, $3 \times 3$, that is symmetric with zero diagonal, I calculate a matrix V, $3 \times 3$, whose columns are the corresponding right eigenvectors and a diagonal matrix D, $3 \times ...
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1answer
22 views

Computation of eigenvectors?

Given a matrix: $A = \begin{pmatrix} -\epsilon & tf_1 \\ tf_2 & -\epsilon \end{pmatrix}$ Compute the eigenvectors. I can easily find the eigenvalues to be $\lambda = -\epsilon \pm t\sqrt{f_1 ...
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1answer
43 views

Basis of the space of linear maps

I asked someone about this problem: Let $V,W$ be vector spaces with bases $(\alpha_i)_{i\in I}, (\beta_j)_{j \in J}$ respectively. Define $f_{ij}(\alpha_k) = \delta_{ik}\beta_j$. Show that ...
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1answer
24 views

Proof of a tr property

$Tr(XY) = 1$ and $Tr(Y) = 1$ implies that $Tr(X) = 1$. I tried to prove by contradiction and switch the dummy variable of $X$ and $Y$. But I don't think my approach is right and if there is any much ...
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1answer
16 views

Given a basis $U$, what conditions are needed for an orthogonal basis for it?

Given a basis $U$, what conditions are needed for an orthogonal basis for it? For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an ...
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If $\operatorname{rank}A=k$ then $A=A_1+…+A_k$ such that $\operatorname{rank}A_i=1$ [on hold]

Let $A\in M_n$ and $\operatorname{rank}A=k$. Is the following true? There are $A_i\in M_n$ ($i=1,...,k$), such that $\operatorname{rank}A_i=1$ and $A=A_1+....+A_k$.
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3answers
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Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$

Let $N\in \text{Mat}(10 \times 10,\mathbb{C})$ be nilpotent. Furthermore let $\text{dim} \ker N =3 $, $\text{dim} \ker N^2=6$ and $\text{dim} \ker N^3=7$. What is the Jordan Normal Form? The only ...
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how to normalise these values

First of all, i don't know if the correct word is normalise or not, but I'll try to explain my issue. I have a relationship between an object A and an object ...
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What's the difference between the trajectory, the phase portrait and vector field of a matrix? [on hold]

Take the matrix $$\begin{pmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{3}{4} & \frac{1}{4} \end{pmatrix}$$ as an example. What's the difference between its trajectory(discrete), phase portrait ...
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matrix trace bilinear form

I'm wondering about this problem on bilinear forms : We have $\phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$ $$(A,B) \rightarrow trace(AB)$$ I've proved $\phi$ is a bilinear form ...
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$SU(n)$ generators

What is the generalization of the Pauli matrices and Dirac matrices in higher dimensions? I am actually looking for $\sqrt{\mathbb{I}}$ but I can't use the principal root which is just $\mathbb{I}$. ...
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1answer
25 views

Find orthonormal basis of quadratic form

Q: Let $$A = \begin{pmatrix} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{pmatrix}$$ Find the quadratic form of $q: \mathbb{R}^3 \to \mathbb{R}^3$ represented by A. and find ...
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34 views

Getting The Inverse Of A Positive Definite Matrix By Mutiplying It On A Diagonal One

Is the following true ? The inverse of a positive definite matrix is also positive definite and since symmetric then we could write the following: $A=PP^T, \space B = A^{-1} = P\Lambda P^T$ since ...
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1answer
35 views

If $V_1$ and $V_2$ are vector spaces over $\mathbb{Q}$ and f is a map $f(x+y)=f(x)+f(y)$ for all $x, y$ in $V_1$. Is $f$ linear transformation?

If $V_1$ and $V_2$ are vector spaces over the field of rational numbers and $f$ is a map from $V_1$ to $V_2$ such that $f(x+y)=f(x)+f(y)$ for all $x, y$ in $V_1$ show that $f$ is a linear ...
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2answers
29 views

Meaning of the phrase “span of vectors contains”

What does it mean if someone says the "span of vectors" $\{(x,y,z),(a,b,c)\}$ contains the vector $(d,e,f)$? I am making up numbers here. Let me know if I need to insert actual numbers here to ...
3
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2answers
50 views

Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$

I need some help figuring out how to work through this problem. Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$. Sorry, forgot to ...
4
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4answers
284 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = ...
2
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2answers
20 views

Minimize Energy Function

Let $A\in\mathbb{R}^{n\times n}$ be symmetric positive definite matrix and $\mathbf{b}\in\mathbb{R}^n$. How to prove that $A\mathbf{u}=\mathbf{b}$ if and only if $\mathbf{u}$ minimizes the so-called ...
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1answer
22 views

Existence of a block upper triangular form matrix representation for a linear operator

Let $T:V \rightarrow V$ be a linear operator on a finite dimensional vector space over $F$. Let $W \subset V$ be a subspace which is $T$-invariant. Show that there exists an ordered basis ...
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1answer
19 views

Is this the (a?) correct definition for $X$ having full row rank?

Let $X$ denote a $T\times K$ matrix. I have seen the definition for full column rank as "There is no vector $c \not = 0$ with $X\cdot c = 0$. Would a definition for full row rank then be "There does ...
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1answer
50 views

Solve this system of equations without calculator

$$2a +4b +3c +5d +6e=37$$ $$4a +8b +7c +5d +2e=74$$ $$-2a -4b +3c +4d -5e=20$$ $$a +2b +2c -d +2e=26$$ $$5a -10b +4c +6d +4e=24$$ find $a,b,c,d,e$ I tried solving the system of equations above but ...