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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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 ...
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34 views

Find a,b,c to match the linear transformation matrix?

P.S. Sorry for my bad explanation of the task, it was really hard to translate this into meaningful english For the given linear-transformation $A$ find all possible combinations of a,b,c for which ...
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1answer
46 views

Is L a linear transformation?

I have to prove is L is a linear transformation on the field $P_3(R)$, if it is then I'd have to find the matrix of the linear transformation from the standard base vectors $p(1),p(x),p(x^2),p(x^3)$. ...
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1answer
10 views

How to find the linear application given the function on the basis vectors.

Say I am given a linear application $f$ from $R^2$ to $R^2$, and I am told it maps $e_1$ to $(1,3)$ and $e_2$ to $(-2,7)$ In this case I know how to find how the linear application acts on a generic ...
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20 views

Solving a generalized eigenvalue problem with constraints

I have the following generalized eigenvalue problem: $ \begin{pmatrix} 0 & a \\ a^T & B / \lambda_{i} \end{pmatrix} \begin{pmatrix} 1 / \lambda_{i} \\ x_{i} \end{pmatrix} =\epsilon_i \begin{...
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36 views

Expectation of Gaussian Ratios

Consider the following expression: $z = \frac{\mathbf{x}^H P \mathbf{x}}{\mathbf{x}^H \mathbf{y}}$, where $\mathbf{y}$ is fixed (not random) and $\mathbf{x}$ is a complex Gaussian vector of zero mean ...
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2answers
44 views

Whats the relation between the set of vector in a vector space to the field that this vector space is over?

We have a set of "vectors" (elements) in the set $V$ such that $(V,+,\circ)$ is said to be a vector space over some field $\mathbb{F}$. Let $F$ be the set of elements that consist the field $\mathbb{F}...
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37 views

Hyperdeterminant of 4x4x4 hypermatrix

If given the hypermatrix (which I've written here in bracket notation since I'm not all too sure how to display this) { {{1,1,1,1},{1,1,-1,-1},{1,-1,-1,1},{1,-1,1,-1}}, {{1,1,1,1},{1,1,-1,-1},{1,-1,1,-...
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15 views

Problem in finding introductory material (matrix spectra)

I am looking for introductory material on: 1) matrix eigenvalue spectra and useful matrix algebra theorems that can be applied in the field. 2) Statistics of random matrices (i.e. ensembles, ...
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34 views

Condition number of a $2\times 2$ square block matrix

Is there a general rule to relate the condition number of the $2\times2$ square block matrix $ \left(\begin{array}{cc} A & B\\ C & D\\ \end{array}\right), $ where the matrices have the ...
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1answer
53 views

Active and passive transformations in Linear Algebra

I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ...
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42 views

What is name of a matrix which construct from a identity matrix?

In linear algebra, the identity matrix ,$I$, or sometimes ambiguously called a unit matrix, of size $n$ is the $n \times n$ square matrix with ones on the main diagonal and zeros elsewhere, for ...
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1answer
72 views

$\text{det}(I-AB)=\text{det}(I-BA)$ [duplicate]

I want to show that for $n\times n$ matrices $A,B$ that $\text{det}(I-AB)=\text{det}(I-BA)$. My thought is that since $AB$ and $BA$ have the same eigenvalues, I know they have they same minimal ...
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1answer
40 views

Ellipse and horizontal lines

Let's imagine that we have an ellipse described by the known equation $v^TAv=0$, (Link_1) where $v=[x \ y \ 1]^T$ (it can be a skew one in a general case). Then we have all possible horizontal lines ...
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12 views

Derive the Hat Matrix to map actual response to estimated resposne

In order to measure the quality of a regression we can calculate the Hat Matrix. Using it we can estimate the response variable as if we used the predictor variables to regress them. For linear ...
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13 views

Projecting down onto a subspace, with weights

I'm following along in an astronomy paper, detailing how to project data with uncertainties and missing values onto pre-computed principal components, and I'm trying to come up with a matrix ...
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19 views

Fourier Analysis for Derandomization of Functions

I was wondering if there was an extension to Fourier Analysis on Boolean Functions. Specifically, it's well known that for any boolean function $$f: \{-1,1\}^{n} \rightarrow [-1,1] $$ we can decompose ...
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1answer
37 views

How does this Tensor Product basis example is made up?

Here is a link for the article. On page 5 of 14 author is talking about tensor product befor that he explains direct sum and does it very clearly by stacking vectors on top of each over. But here (the ...
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2answers
23 views

general solution for a linear map

If $L$ is a linear map $L : V \rightarrow W$, and $L(v) = w \in im L$, show that $$L^{-1}(w) = v + ker L$$ ('im L' is the image of L, and ker L is the kernel of L) I tried it by using the ...
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1answer
19 views

Dimension of $MS$ where $M\in M_{m\times n}$ and $S\le \mathbb{R}^n$

Could anyone tell me how to find the formulae for $\dim MS$ where $M\in M_{m\times n}(\mathbb{R})$ and $S\le \mathbb{R}^n$ be a subspace and $MS=\{Mx:x\in S\}$ I thought $M:\mathbb{R}^n\to \...
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1answer
29 views

Null space isomorphic to range $T$ [on hold]

Let $V$ and $W$ be vector spaces over $K$ and let $T:V \to W$ a linear application. Prove that $V/(\text { null} (T))$ is isomophic to range of $T$.
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50 views

rank of the matrices of the form $A+\lambda B$

Could anyone help me for the following problems? $1. $ If $A+\lambda B\in M_{m\times n}(\mathbb{R})$ where we can vary $\lambda\in \mathbb{R}$, has rank $r<n$ then what can we say about the rank ...
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16 views

Mean value theorem for a gradient of convex function

This is from an article, page 19. Let $J(u)=\sum \sqrt {u_i^2+\epsilon}$, and $p^{k+1}=\nabla J(u^{k+1})$, $p^{k}=\nabla J(u^{k})$. Since $J$ is convex, the mean value theorem tells us that $$p^{k+...
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34 views

Resolvent Inequality

Let $H$ be a Hermitian matrix and $h$ some vector of the same length. The resolvent of $H$ at $z\in\mathbb C$ shall be denoted by $$G(z):=(H-z\cdot1)^{-1}.$$ Is it true that $$\frac{(\Im z)\left(1+\...
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1answer
28 views

Notation for set of unit vectors

Is there a standard notation for the set of unit vectors $\{\vec v\ :\ |\vec v|=1\}$?
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29 views

If I have n different eigenvalues prove their eigenvectors are linealy independent [duplicate]

Prove via induction that if $V$ is a vector space of finite dimension and T: $V\to V$ a linear operator with n different eigenvalues then the eigenvectors associated with them are linearly independent....
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29 views

Direct sum of matrices

Let $\mathbb{M}_n(K)$the set pf all $n\times n$ square matrices with coefficients in $K$. And let $V$ the set of all lower triangular matrices, and $U$ the set of all upper triangular matrices with ...
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21 views

Find rotation matrix to match points in parallel projection

I am given two sets of 3D points (actually 2D, see below) with corresponding pairs. I am seeking two 3D rotation matrices, such that (only) the X and Y components of the rotated points match best (...
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1answer
44 views

In the context of linear algebra, is it possible for a vector space or a subspace to have a finite number of elements? [duplicate]

A vector space must satisfy closure under addition and multiplication. Sorry if this is obvious but does that mean that, assuming the normal rules of arithmetic and excluding the trivial examples like ...
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1answer
53 views

Rank of the matrix

Let ${\bf A} $ is a matrix that construct with coordinate of the $n$ distributed points in a two dimensional domain like follow: $${\bf A }=\begin{pmatrix} 1&x_1&y_1\\ 1&x_2&y_2\\ \...
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Solve $\max \mathrm{sum}(AXB \geq \gamma), X \in \{0,1\}^{N \times N}$

I have a problem to find the best permutation matrix $X \in \{0,1\}^{N \times N}$, so as to maximize the number of elements in $AXB$ which are above a certain positive number $\gamma$. In other ...
4
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1answer
65 views

Converse of Schur's Lemma in finite dimensional vector spaces

I am trying to prove (or disprove) the converse of Schur's Lemma in finite dimensional vector spaces. I am not sure if it holds in this case, but I have tried to apply the idea that proves it in ...
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45 views
+50

Moore-Penrose pseudoinverse and Linear relations

I recently came across this website called Graphical Linear Algebra. I feel like there's a lot of insight there, but it's too monolithic for me to be able to extract it by skimming. Episode 27 is ...
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If $Y = X\beta$ are a system of linear equations and that $X$ is NOT full rank. Is this system under or over determined?

Suppose I have a system of linear equations, $Y = X\beta$, where $Y$ is a $n$ by $1$ matrix, $X$ an $n$ by $n$ matrix, and $\beta$ a $n$ by $1$ matrix. Suppose that I know what $Y$ and $X$ are, and ...
2
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1answer
18 views

Can we get $\|A^\dagger x-B^{-1}x\|_2\leq \epsilon \|B^{-1}x\|_2$?

In the question: the $A\in R^{d\times d}$ is positive semi-definite, $B\in R^{d\times d}$ is positive definite, $x\in R^d$ is a vector, and $\epsilon$ is a variable that may depend on $A$, $B$ and ...
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1answer
17 views

Can a scalar multiple for a linearly dependant vector be undefined?

Let $A = \begin{bmatrix}a & b & c \\ 2a & 2b & 2c \\ e & f & g\end{bmatrix}$ where $R_{3}$ is linearly independent of $R_{1}$, Clearly, $(-2 , 1, 0)$ will be set of scalars ...
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22 views

Show that this vector is not a function of $\tau$

I have a variance matrix given by: $\boldsymbol{\Sigma}\boldsymbol{\Sigma}^{'}+\Omega$ where $\Omega=\left(\begin{array}{cccc} \sigma_{\varepsilon}^{2}\psi\left(\tau_{1}\right) & 0 & \...
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1answer
64 views

Spectral theorem for diagonlizable matrices

For a diagonalizable matrix $\textbf A_{n \times n}$ with spectrum $σ(\textbf A)=\{\lambda_1, \lambda_2,..., \lambda_k\}$ we have matrices $\{ \textbf G_1, \textbf G_2,..., \textbf G_k \}$ such that: ...
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0answers
27 views

Why does power iteration generate almost dependent vectors?

On the Wiki page for Krylov subspaces: https://en.wikipedia.org/wiki/Krylov_subspace it states given a matrix $A$ and vector $b$, that the vectors $b, Ab, A^2b, A^3b, ...$ "soon become almost linearly ...
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3answers
31 views

Solve for $X$ in matrix equation

How can I solve for $X$ in this matrix equation? $$\begin{bmatrix}-3&-8\\-9&5\end{bmatrix} X + \begin{bmatrix}4&-7\\3&-2\end{bmatrix} = \begin{bmatrix}5&8\\-1&-1\end{bmatrix} ...
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0answers
32 views

Inner-product on skew-hermitian matrices

Let $$\mathfrak{u}(n)=\{X\in M(n,\Bbb C):X+X^*=0\}$$ where $X^*$ is the conjugate transpose. Then, $\mathfrak{u}(n)$ is a real vector space. Problem. Show that $\langle X,Y\rangle=\...
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Example of a degenerate bilinear map?

I seek an example of a nonzero $\Bbb{R}$-bilinear map $f:V\times V\rightarrow W$ on a vector space $V$ (s.t: $\dim V<\infty$, $\dim W<\infty$) such that it is degenerate map, where $V$ and $W$ ...
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24 views

How to estimate the product of the $k$ largest eigenvalues of a matrix

Now I have a question which let me to prove that the product of the largest $k$ singular values of a real matrix is always larger than the one of $k$ largest eigenvalues. For $k=1$, I use the ...
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Spectral norm of the matrix derivation

I understand one possible way how to derive induced norm of symmetrix matrix M, i.e. $sup |M \tilde{x} |$, s.t. $|\tilde{x}|=\tilde{x}^T\tilde{x}=1$ (i.e. $\tilde{x}$ is lie in unit sphere) Here is ...
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24 views

Dimension from inequalities

Let $S$ denote $(x,y,z)\in \mathbb R^3$, which satisfies the inequalities: $$x - 2y + z \leq 1$$ $$2x + 2y - z \leq 5$$ $$-2x + y + z \leq 4$$ $$x \geq 1$$ $$y \geq 2$$ $$z \geq 3$$ How do I ...
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14 views

Using Fixed point iterations for solving system of linear equations

Given a system of $n$ linear equations $$ x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find $x_i$. The fixed point iteration define $$ x_i^...
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4answers
34 views

whether set of 3D vectors span space {(x, y, z) | x + y + z = 0}

Consider the set of (column) vectors defined by $X = \{x \in R^{3} | x_{1} + x_{2} + x_{3} = 0\}$, where $X^{T} = [x_{1}, x_{2}, x_{3}]^{T}$ , I need to prove whether(or not) given vectors, $[1, -1, 0]...
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15 views

Solving $I^* = \arg\min_{I'} \left( \|\phi_\ell(I) - \phi_\ell(I')\|_2^2 + R(I') \right)$ with gradient descent

I am trying to create the results from this a paper that is trying to understand the types of features a convolutional neural network is learning to recognize. I don't think understanding ...
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2answers
30 views

Given the matrix, find a matrix such that

Given $T(\begin{bmatrix}1\\-2\end{bmatrix}) = \begin{bmatrix}3\\10\end{bmatrix}$ $T(\begin{bmatrix}-2\\-1\end{bmatrix}) = \begin{bmatrix}-1\\-5\end{bmatrix}$ Find a matrix such that: $T(...
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2answers
24 views

how to know if a set of arbitrary vectors are a basis?

So, if we're given that $\{x,y,z,w\}$ is a basis of $\Bbb R^4$, how do we show that $\{x+w, y+w, z+w, w\}$ is also a basis of $\Bbb R^4$? I know that for a set to be a basis, it has to be linearly ...