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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Why is the following number always positive?

consider two points in the Euclidean plane $A=(A_1,A_2),B=(B_1,B_2)\in\mathbb{R}^2$ and some fixed real number $\lambda\in(0,1)$. The claim is that the following expression is always a positive real ...
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19 views

Matrices and basic properties

Let $A$ be an invertible 10x10 matrix with real entries such that the sum of each row is $1$. Then choose the correct option. The sum of the entries of each row of the inverse of $A$ is $1$. The sum ...
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2answers
17 views

An invertible sparse matrix?

I'm not entirely certain about how to tackle this problem.... I hope you ladies and gents can help :) If $M\in M_{n\times n}(\mathbb{R})$ be such that every row has precisely tow non-zero entries, ...
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24 views

Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F )$.

Let $F$ be a field and let $n$ be a positive integer. Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F )$. If I set $A=I$, ...
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1answer
48 views

Why multiplying those 2 quaternions doesn't give the expected result?

Using a left-handed coordinate system, let Q = axisAngle({0,0,1}, 1/4*pi) * axisAngle({0,1,0}, 1/4*pi) be the quaternion representing the rotation "1/8 circle ...
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1answer
21 views

Show that any nontrivial ideal that is also a subspace of $M_{n\times n}$ over a field $F$ is maximal.

I just need to show that if $A$ is a nonzero square matrix, then I can "build" the identity matrix $I$, Then the ideal must be maximal.
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1answer
36 views

Linear Transformations in Linear Algebra

We are given: Show how to evaluate a linear transformation for a specific vector $x$ , when the transformation is defined in the form $$T(x) = y$$ We know that a linear transformation is defined as ...
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21 views

functions (on intervals) in vector spaces

I'm studying mathematical methods for solving physics and engineering problems. I've looked in a few books and I'm curious about functions being manipulated like vectors. How could I find out more ...
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21 views

Solving System of Equations modulo a prime

Consider the equation: $$ C \equiv H.M.H^{-1} \pmod{p}, $$ where C, M H are, say, $2\times 2$ matrices, and $p$ is an odd prime. The elements of the matrices $C, M$ are all integers. The elements ...
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34 views

Prove a matrix is non-negative.

Let $\textbf{r}_1$ and $\textbf{r}_2$ be $n \times n$ symmetric, diagonally dominant, Metzler matrix with eigenvalue $\max(|\lambda_i|)<1$ for both $\textbf{r}_1$ and $\textbf{r}_2$. Let ...
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6 views

Comparison between interpolation and Tikhonov regularization.

Interpolation is defined as finding a value of a function between two points and one can think of Tikhonov regularization as to estimate a suitable function under certain condition. Can we think ...
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1answer
20 views

Eigenvectors times diagonal matrix, still eigenvectors?

Suppose we have a $n\times n$ real symmetric positive definite matrix $\Sigma$, and $V=(v_1,...,v_n)$ whose columns are the eigenvectors corresponding to the $n$ eigenvalues $\lambda_1\geq \lambda_2 ...
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11 views

QR and Cholesky decomposition

A while ago I asked for help to develop a polynomial regression model using least squares, where the system was solved by cholesky decomposition, you can check it here Cholesky Polynomial Regression ...
2
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1answer
24 views

Singular matrix with entries in a ring.

Given a matrix $M\in A^{n\times n}$, where $A$ is a commutative ring different from $\{0\}$, then we know that if there exists a vector $x\in A^n$ such that $Mx=0$, then $\det M$ must be a zero ...
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18 views

Isomorphism among quotient algebras

Let we have two different quotient algebras $ A/B$ and $M/N$ such that $B$ and $N$ are nilpotent ideals of $ A$ and $M$. I want to define an isomorphism $ \phi: A/B \to M/N$.Can you help me how I ...
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27 views

Need a formula / method to get a value between 0 and 1 if a point lies in an area between two rectangles

I'm trying to figure out a way to construct a formula or method for the following process: If Point E is inside the smaller rectangle (red area), the value should be 1. If Point E is outside the ...
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22 views

Proof that $\text{span} \{v_1,…,v_k\} \cap \text{ker}(T) = \{0\}$ if $\{v_1,…,v_k\}$ are vectors in general linear position.

The problem set up is as follows: Let $\omega^{(i)} \in \mathbb{R}^n$, for $i=1,2,...,k$, $k \le n$, be i.i.d. random vectors (whose distribution is irrelevant). Also, let $A \in \mathbb{R}^{m \times ...
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28 views

Orthogonal projection af a $5\times3$ matrix onto a subspace spanned by two of its vectors.

As a part of a data analysis exercises I need to project a matrix that contains $5$ observations of $3$ variables onto a plane spanned by two of those variables. I can't really imagine this. What is ...
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1answer
58 views

Proving that every vector space has a norm.

I am trying to prove that every vector space $X$ has a norm. I have some silly questions, but it's better to ask them now instead of later. I think I'm having a bit of trouble getting intuition about ...
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2answers
18 views

Complex inner product linearity

Let $V$ be an inner product space over $\mathbb{C}$. Is the expression $$ \newcommand{\<}{\langle} \newcommand{\>}{\rangle} \<v,\lambda u\> = \bar{\lambda}\<v,u\> = ...
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1answer
28 views

$\overline{\mathrm{Im} (T^*T)} = \overline{\mathrm{Im} T^*}$

I need to prove that in a Hilbert space, $\overline{\mathrm{Im}(T^*T)} = \overline{\mathrm{Im}T^*}$. I have already shown that $\ker (T^*) = (\mathrm{Im} T)^\perp$ and have so far concluded that ...
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44 views

Distinct eigenvalues of matrix

While studing Quantum Physics I've came to the conclusion that it would be useful to find an equivalent (or at least a sufficient) condition for a matrix (over $\mathbb{C}$) to have distinct ...
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Linear Operators, Kernel and Orthogonal/Unitarily Diagonalization

Could someone please help me? Here is the question: Given the linear operator $T:C^3 \rightarrow C^3$, whose standard matrix is $$ \left[ \matrix { 1&0&0 \\ x&1&0 \\ 0&1&1 ...
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3answers
35 views

Name of Jordan Canonical Form in infinite dimensions?

I tend to think of Jordan canonical form as the generalized spectrum theorem. I read it as saying, every matrix cannot be diagonalized, but they can be "jordanized". In functional, I've seen the ...
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2answers
43 views

Subspaces dimensions in $\mathbb{R}^7$

if $U$ and $W$ are subspaces of $\mathbb{R}^7$ and $\dim U = \dim W =4$ then in $U \cap W$ there's a vector different then $0$. I think that it's true, am I correct?
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2answers
36 views

Finding eigen values and eigen vectors of the linear transformation.

I have a doubt that can we find Eigen values and eigen vectors for the linear transformation, I only know to transform linear transformation into matrix form using standard basis or any other basis ...
2
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2answers
19 views

Coordinate vector equation

I have the following bases which are bases of $\mathbb{R}^3$ $$B = ((1,1,1), (0,1,1), (0,0,1))$$ $$C = ((1,2,3), (-1,0,1), (1,0,1))$$ I need to find if this equation is correct $$[(1,2,3)]_B = ...
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1answer
20 views

using matrix with cos/sin etc.

I need to check if the equation is linear independent so: $$ \alpha x^2 \cos x + \beta x + \gamma \sin x = 0 $$ I got 3 equations of it: $$\beta \pi/2 + \gamma = 0$$ $$\alpha \pi^2(-1) + \beta \pi = ...
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2answers
40 views

If I have a matrix M=[A,B;0,C], how do I prove that rank(A)+rank(C)<=rank(M)?

. . . . . . . A . . B . . . . . . . 0 0 0 . . . 0 . 0 . C . 0 0 0 . . . If I have a matrix $M$ as displayed in the text above ($A$ ...
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7answers
566 views

What is the idea behind a projection operator? What does it do?

I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can anyone help me? I don't need examples or the definition - I want to know why and ...
5
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2answers
216 views

Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the ...
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0answers
21 views

Projections onto a subspace (orthogonal vs. non-orthogonal matrix vs. basis matrix)

Suppose $A$ is our matrix $B$ is our basis for the matrix $A$ $Q$ is orthogonal basis for matrix $A$ $P=A(A^TA)^{-1}A^T$ Is the following true: $Px=(QQ^T)x=(B(B^TB)^{-1}B^T)x$ ...
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1answer
58 views

$A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?

Suppose $A,B \in {M_n}$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?
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0answers
20 views

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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14 views

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
58 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
22 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
29 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
24 views

linear relations - algebra [on hold]

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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14 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
68 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
28 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 ...
2
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0answers
25 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
29 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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49 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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1answer
29 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
36 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
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3answers
45 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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23 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 ...