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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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
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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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12 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
7 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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25 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
29 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 ...
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38 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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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
19 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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37 views

What does it mean for a function to be closed under linear operators?

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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28 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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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
20 views

Tensor contraction

Given that: $T_{i,j}=\lambda\theta\delta_{i,j} + 2\mu E_{i,j}$ Show that: $T_{i,i} = 3\lambda + 2\mu E_{i,i}$ I didn't get the intuition behind tensor contraction, thus i can not solve this ...
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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
22 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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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
11 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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17 views

Random projection onto orthonormal bases

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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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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2answers
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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
17 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 ...
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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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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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21 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
17 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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33 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
18 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
29 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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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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15 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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Orthogonal Projections Composition iff Statement

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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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
25 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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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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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
206 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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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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37 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) $W = \{x = (x_1 , x_2 ...
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$A^2=A^*A$.Why $A$ is Hermitian matrix?

Let $A$ be $n \times n$ matrix and $A^2=A^*A$. Why is $A$ a Hermitian matrix?
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35 views

Spectrum of the matrix $A=(a_{ij})$ where $a_{ij}=i+j$

What is the spectrum of the matrix $A=(a_{ij})_{n\times n}$ where $a_{ij}=i+j$ for any $n$. Also, what are the eigenvectors corresponding to their eigenvalues? Progress. This matrix is definitely ...
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A question about the vector space spanned by shifts of a given function

Let $E$ be the space of continuous real functions and $f\in E$ Let $T_t$ denote the shift operator: $T_t(f)(x)=f(t+x)$ Let $T(f)$ be the linear span of the set $\{T_t(f) \;|\; t\in ...
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1answer
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Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} $?

Let $A \in {M_n}$ be hermitian and suppose that at least one eigenvalue of $A$ is positive ($\lambda $ is eigenvalue of $A$). Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} ...
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31 views

Second order derivation of Quadratic form

I would like to find the second order derivative of a Quadratic form. Assume we have a random complex column vector $x$ and a real constant value $C$. I am interested in computing the following: $$ ...
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Is there an easier way to show that a kite has perpendicular diagonals using scalar products?

I want to use scalar products to prove that a kite has perpendicular diagonals. My attempt : Let $a,b,c,d$ vectors with $a+b+c+d=0$ and $a^2=d^2$ and $b^2=c^2$ Then, we get ...
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Number of two dimensional sub spaces of a vector space over a finite field.

Let {$e_1,e_2,e_3,e_4$} br a basis of $4$-dimensional vector space over a finite field with p elements. The number of $2$-dimensional subspaces of $V$ not containing $e_4$ and not contained in ...
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Circulant matrix - Eigen decomposition in matlab [on hold]

$A = FD{F^{H}}$ , where $A$ is a circulant matrix, $F$ is normalized FFT matrix (unitary) and $F^{H}$ is the conjugate transpose (Hermitian) of the $F$ matrix. $D$ is the diagonal matrix which ...
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Linear Algebra L2 minimization

Im really confused about how to solve this question or even what its asking. Any help would be much appreciated! Let A be an m x n real matrix ($m \gt n$). Let x* be the minimizer of $||Ax - b||^2 + ...
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40 views

Help understanding the range and kernel of a linear transformation

I'm having some trouble understanding the Range and Kernel of a linear transformation. The definition goes as follows: Let $T:V \longrightarrow W$ be a linear transformation. Define the sets ...