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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Equivalence class of matrices on linear form

We well know that if $M$ is a matrix on a field $k$ then the equivalence class of $M$ is uniquely determined by its rank (where $A \sim B$ if $\exists P,Q $ invertibles such that $PAQ^{-1}=B$). ...
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0answers
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Dependence of isomorphism of two vector spaces with Their fields

V and G be two vector spaces over field F and G respectively. Can they be isomorphic if F is not isomorphic to G? What if F is isomorphic to G? I've no idea where to start...Please help
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2answers
49 views

Why is the following set not a vector space?

I have the following set and I want to know whether it's a vector space or not: $W = \{(x, y, z) ∈ \Bbb R^3 : (x + y)(2y − z) = 0\}$ Now, I understand that if I have a set W and it's a vector ...
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1answer
19 views

Invertibility of character table

Corollary. The character table of a group is an invertible square matrix. The theorem that is a corollary to states that the character table is a square matrix and the explanation for invertibility ...
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0answers
24 views

About an inequality

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be two sets of n real numbers. Say that there is at least one element different between the two sets. Can one estimate for how large a positive ...
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28 views

Why does $rank(A) \ge \dfrac{{{{(trA)}^2}}}{{(tr{A^2})}}$? [duplicate]

Let $A \in {M_n}$ and Hermitian.Why does $rank(A) \ge \dfrac{{{{(trA)}^2}}}{{(tr{A^2})}}$?
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1answer
18 views

linear relations - algebra

A baker makes a loss of $30$ when $25$ cakes are sold but makes a profit of $100$ when $90$ cakes are sold. What is the linear relation for his profit? (Please show me the steps; I can do many other ...
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0answers
6 views

Holder's inequality/Cauchy-Schwartz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) ...
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3answers
61 views

Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$ [on hold]

Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$. How to find its numerical range $W(A) = \{ {x^*}Ax:x \in {S^1}\}$?
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2answers
25 views

Inequality involving inner product and an othonormal set of vectors

$ \newcommand{\ip}[2]{\left\langle #1,#2 \right\rangle} $ Here is the statement of the problem: Suppose that $V$ is a real inner product space with an inner product $\langle\cdot,\cdot\rangle$, and ...
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0answers
22 views

Closed form of a matrix product

Is there any closed form or a bound for a matrix product of this kind $$ P=\prod_{i=1}^n \begin{pmatrix} 1-a & a \\ b_i & 1-b_i \end{pmatrix}, \quad a,b_i \in [0,1] $$ for an arbitrary ...
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2answers
28 views

Remembering the definition of the Jacobian: any tips?

I find it impossible to remember that the Jacobian of $f: \mathbb R^n \to \mathbb R^m$ is $$ \begin{pmatrix} {\partial f_1 \over \partial x_1} & {\partial f_1 \over \partial x_2} & \dots ...
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0answers
38 views

Help with Linear Algebra Optimization Problem. 4 people crossing a bridge

"Four people, A, B C and D need to get cross a river. The only way to cross the river is by an old bridge, which holds at most 2 people at a time. Being in dark, they can not cross the bridge ...
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1answer
10 views

Show that a vector can be represented in term of its components

How do i prove this identity: $\vec{a} = a_1\vec{e_1} + a_2\vec{e_2} + a_3\vec{e_3} = a_i\vec{e_i}$ $\vec{e_i}$ are the unit vectors For instance: $(1,0,0), (0,1,0), (0,0,1)$ if we have three ...
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2answers
28 views

What's the difference between linear span and linear transformation?

I tried to google both definitions. For linear span, click http://en.wikipedia.org/wiki/Linear_span For linear transformation(wiki takes it as linear map), click ...
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3answers
35 views

How to find the basis of the following vector space?

I'm trying to find the basis of the following vector space but I can't seem to be able to find it: $W = \{x = (x_1 , x_2 , x_3 , x_4 , x_5 ) ∈ \Bbb R^5 : x_1 − x_3 − x_4 = 0\}$ I understand that ...
2
votes
3answers
43 views

Are the coefficients of a vector according to a basis unique?

If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as ...
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0answers
18 views

Comprehensive easy to understand resource for learning matrix decompositions?

I am working on my thesis which is widely depended on knowledge about matrix decompositions. I have studied linear algebra with the help of YouTube videos, MITOpenCourseWare videos and Prof.Gilbert ...
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1answer
23 views

Show that a field extension $L/K$ is separable iff the trace form is non-degenerate.

Let $L$ be a finite field extension of $K$. I have the following question: Show that $L/K$ is separable if and only if the bilinear trace form $\text{Tr}_{L/K}:L\times L \to K$ is non-degenerate. ...
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1answer
43 views

What does it mean for a function to be closed under linear operators? [on hold]

What does it mean for a function to be closed under linear operators? I'm looking for as informal and intuitive of an explanation as possible.
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2answers
37 views

In euclidean space, $\forall x\in E:\|f(x)\|\le \|x\|$ implies $\ker(f-id)\oplus \mathrm{Im}(f-id)=E$

Le $E$ be an euclidean space, $f\in\mathscr L(E)$, such as $\forall x\in E:\|f(x)\|\le \|x\|$. Show that $\ker(f-id)\oplus \mathrm{Im}(f-id)=E$. I've tried to show that $\ker(f-id)\perp ...
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0answers
14 views

isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
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1answer
25 views

Tensor contraction

Given that: $T_{i,j}=\lambda\theta\delta_{i,j} + 2\mu E_{i,j}$ Show that: $T_{i,i} = 3\theta \lambda + 2\mu E_{i,i}$ I didn't get the intuition behind tensor contraction, thus i can not solve this ...
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0answers
27 views

augmented matrix question [on hold]

Please show me the augmented matrix solution with steps for the system $$ \begin{cases} 3x + y+z=18 \\ 4x + 2y+3z=12 \\ 7x + 8y+5z=9 \end{cases} $$
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1answer
24 views

Finding the property of a basis

Let $V = P_2 [x]$, the vector space of polynomials of degree at most 2. Given that $\mathcal B \subset V$, I want to find whether the following is a basis, not linearly independent, not spanning, or ...
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1answer
22 views

Taking derivative of $\log \det[x(I - y H)^{-1} + z I] \text{ w.r.t. }x, y, z$

Let $C$ be an $n\times n$ symmetric positive definite matrix, and $H$ be an $n\times n$ symmetric matrix. Let $$ f(x, y, z) = \log \det[x(I - y H)^{-1} + z I]. $$ Is there any explicit formula for ...
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1answer
13 views

Finding an ordered basis to diagonalize Transpose matrix.

We define $T : M_{n \times n}R \to M_{n\times n}R$ by $T(A) = A^t$. We can write the matrix representation of this transformation as: $[T]_\beta^\beta = \begin{pmatrix} ...
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0answers
18 views

Random projection onto orthonormal bases [on hold]

Given an arbitrary N dimensional vector of length $L$, and a $M$ dimensional orthonormal basis chosen uniformly at random with $M<N$, what is the CDF of the length of the projected vector?
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3answers
31 views

For which values of $a$ does the matrix can be diagnolized?

Given $$A=\begin{pmatrix} 2 & 0 & 0\\ a & 2& 0\\ a+3 & a &-1 \end{pmatrix}$$ For which values of $a$ can $A$ be diagonal? I found that $p_A(x)=(x-2)^2(x+1)$ and tried to ...
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votes
2answers
15 views

Solving inequality(limit)

Can someone explain how we get from $(x - 3) < \varepsilon/8$ and $x < 4$ to: $(x-3)(x+3) < (\varepsilon/8)(4+3) = (7\varepsilon)/8$
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1answer
21 views

Each bilinear form induces a unique bilinear form from the dual space

Let $V$ be a finite dimensional vector space over a field $K$. Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form. I now want to show that there exists one and only one bilinear form ...
3
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3answers
53 views

Why is cross product not commutative?

Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why ...
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0answers
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0
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0answers
11 views

finding equality with subspaces direct sum

assume that $U_1 \cap U = \{0\}$ and $U_2 \cap U = \{0\}$ $U_1 \oplus U = U_2 \oplus U$? I thought that it's correct because I could find a counterexample.
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1answer
23 views

What is $\max(\operatorname{Re} \{ \frac{x^* Ax}{x^* x}:0 \ne x \in C^n\} )$?

Let $A = \left( \begin{array}{*{20}{c}} 1&2\\ 0&1 \end{array} \right)$. What is $\max\left(\operatorname{Re} \left\{ \dfrac{x^* Ax}{x^* x}:0 \ne x \in C^n\right\} \right)$?
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1answer
19 views

Why is the transformation the geometrically projection ? Where and along what is projected?

Find eigenvalues and eigenvectors of the matrix in standard basis corresponding matrix
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3answers
36 views

Linear Dependent Span

$\{x \cos x, x, \cos x \}$ is a subspace of $V$. I need to find if it's a linear dependent or linear independent. So I thought that its dependent since $x \cos x$ is multiplication of $x$ and $\cos ...
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1answer
25 views

Similarity of a specific block matrix

Let $A$ and $B$ be square matrices of same dimension. I considered $n$-by-$n$ block matrices of the form \begin{align*} \begin{pmatrix} A & & \\ & \ddots & \\ & & A ...
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1answer
31 views

What are the facts used in each step of this proof?

What are the facts used in each step of this proof ? Suppose that $A\in F^{nm}$ and $B\in F^{ml}$ $$\begin{align}rank A + rank B &= rank\begin{bmatrix}0 & A\\B & 0\\ \end{bmatrix}\\ ...
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votes
1answer
15 views

an inequality for the projection on the intersection of 2 subspaces

Can someone give an inequality, bounding the distance of a point from its projection on the intersection of two subspaces by a function of both the distances of the point from the insividual ...
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0answers
16 views

Hermitian, orthogonal, unit trace matrix bases.

Consider the vector space of Hermitian matrices acting on a finite dimensional vector space, equipped with the Hilbert-Schmidt norm. I'm interested in matrix bases that satisfy three properties: 1) ...
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0answers
23 views

Orthogonal Projections Composition iff Statement [duplicate]

Given m , n ⊂ V a finite dimensional inner product vector space, Prove that for Pm, Pn orthogonal projections onto m, n accordingly, PmPn is an orthogonal projection ⇔ PmPn = PnPm and in this case ...
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12 views

find Jordan form

Determine the jordan form of $A = \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4 \end{pmatrix} $ First, I find the characteristic polynomial. $C_A(x)=(x-1)(x-4)^2$. ...
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4answers
27 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
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0answers
14 views

upper bound for the sum of trace related to product of two matrices?

Given A and B positive definite matrices The inequality $\sqrt{4tr(AB)}$ $\leq$ $tr(A+B)$ is lower bound for tr(A+B) is there another inequality for the upper bound, i.e. ?? ≥ tr(A+B)?
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2answers
57 views

Findinf the remaining eigenvalues of a $3\times 3$ matrix

Let $n$ be a fixed natural number. We wish to compute the eigenvalues of the matrix below: $$\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$$ We have a condition on the ...
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3answers
209 views

Help determining whether a transformation is linear or not

I have the following transformation: $T : \Bbb R^3 → \Bbb R^2 , T (x_1 , x_2 , x_3 ) = (x_1 − x_2 , 2x_2 )$ I need to determine whether it's a linear transformation or not. I understand that ...
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2answers
33 views

Find the standard matrix for the transformation [on hold]

Find the standard matrix for the transformation that projects $$ \begin{bmatrix} 1\\ 2 \\ \end{bmatrix} \text{ to } \begin{bmatrix}2\\ 4 \\ \end{bmatrix} $$ and $$ \begin{bmatrix} 1\\ 1 \\ ...
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1answer
38 views

Prove that if $A,B\in M_n(\mathbb{F})$ are $(n-1)$-nilpotent then they are similar.

If $A,B\in M_n(\mathbb{F})$ are $n-1$ nilpotent, prove they are similar. Can I say that, since their minimal polynomial is $X^{n-1}$ they are similar? I know that If $A,B$ are similar, they have ...
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3answers
37 views

How to find the basis of the following vector spaces?

I'm trying, in vain, to find the basis of the following vector spaces: (a) $W = \{x = (x_1 , x_2 , x_3 ) ∈ \Bbb R^3 : x_1 − 2x_2 + x_3 = 0, 2x_1 − 3x_2 + x_3 = 0\}$ (b) $W = \{x = (x_1 , x_2 ...