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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linear algebra characteristic values

Let $T$ be the linear operator on $R^4$ which is represented in the standard ordered basis by the matrix $$ \left( \begin{matrix} 0 0 0 0 \\ a 0 0 0 \\ 0 b 0 0 \\ 0 0 c 0 \end{matrix} \right) $$ ...
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2answers
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Matrices where A^2 = A

I have a feeling that the only invertible matrix - A . that when it squared A^2 is still A , is the Identity matrix . Am I right? and if so , could anybody show me the proof?
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4answers
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Change the matrix by multiplying one column by a number.

Consider a positive definite matrix A. We can think of the matrix as a linear transformation. Now supopse we get matrix B by multiplying only one column of A by a number. Is there a geometric relation ...
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What scaled version of vector to use in QR-factorization when vector subtraction is involved

Im trying to figure out if I understand the conceptual basics. All the time you see that vectors are scaled down/up for readability or for simplifying future calculations with that same vector. As ...
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Are these two definitions of an affine subspace equivalent?

I've seen the notion of an affine subspace defined differently as follows: $S \subset \mathbb R^3$, non-empty, is an affine subspace if $(1-t)u + tv \in S$ whenever $u,v \in S$. $S$ is an affine ...
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theorem involving changing bases.

the theorem is as follows: Let A be the matrix of T:U -> V with respect to the bases {e i} of U and a basis {f j}of V, and let B be the matrix of T with respect to the bases {e' i} of U and a basis ...
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0answers
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Maximal ideals of the ring of matrices

Let K be a field. We consider $K^n$ as a left module of $M_{n, n}(K)$, the ring of matrices of size $n$ over $K$. 1) For any $M_{n, n}(K)$ module homomorphism $ 0 ≠ \phi: M_{n, n}(K) \to K^n$, show ...
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1answer
21 views

Subspaces: Does closure under scalar multiplication imply additive identity?

Working through Axler's Linear Algebra Done Right (second edition), I came upon the following assertion: If $U$ is a subset of a vector space $V$, then to check if $U$ is a subspace of $V$ we only ...
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Linear operators proof, projection and reflection matrices

I am trying to understand two parts from the picture below in my textbook, but I dont understand how they arrived at it. I am trying to understand the proof below and how they got $P_L(\vec{v}) = ...
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1answer
15 views

Reduced row echelon form of full rank matrices

Does the row echelon form of a full rank square matrix ALWAYS reduce to identity matrix? Thanks
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1answer
24 views

Finitely generated modules and submodules

Let $R$ be a ring, $M$ an $R$-module and $U$ a submodule of $M$. Show that, if $U$ and $M/U$ are finitely generated, then $M$ is finitely generated aswell. I thought to maybe show this by taking ...
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3answers
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Determine 9 variables by 3 equations with approximation

I have an equation in the form of Q*d=z, where Q is 3by3 matrix of variables, and d and z are vectors of 3 known numbers. What would be the best way to compute all 9 elements of matrix Q, provided ...
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find spectrum matrix A

Let $A = \left[ \begin{array}{*{20}{c}} 0&b&0&0&0&0\\ c&0&b&0&0&0\\ 0&c&0&b&0&0\\ 0&0&c&0&b&0\\ ...
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0answers
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squared trace inequality for hermitian matrices

I was wondering how to proof that $Tr(H^2)\cdot d - Tr(H)^2\geq 0$ for each $(d\times d)$ Hermitian matrix $H$. This is equal to $d\sum_j \lambda_j^2-\sum_{j,k}\lambda_j\lambda_k$ with eigenvalues ...
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11 views

Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
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1answer
32 views

How to find the dimension of the given vector space

Let $L=\{p(B)|\ p\ \text{is a polynomial with real coefficients}\},$ where $B =\begin{pmatrix} 0 & 1 &0\\0 & 0&1\\ 1&0&0\end{pmatrix}.$ Then the dimension $\;d\;$ of the vector ...
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0answers
30 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
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How can I determine if a transformation is onto

Is (x,y) mapped into (x,y,0) an onto transformation? If I use the theorem that the dimension of V is less than the dimension of W, then I think that it's not onto. However, I don't see a vector in W ...
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2answers
37 views

Process of finding the eigenvalues of a 3x3 matrix

I'm trying to find the eigenvalues of a 3x3 matrix in order to eventually find an orthogonal matrix $Q$ and diagonal matrix $D$ such that $Q^TAQ = D$, where $A$ is a symmetric matrix, however I'm not ...
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3answers
123 views

Any hint about solving this monster determinant?

I'm asked to solve the following determinant: $$|A|= \begin{vmatrix} 1 &2 &3 &\cdots &{n-1} &n\\ 2 &3 &4 &\cdots &n &1\\ \vdots &\vdots &\vdots & ...
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How are these definitions of the inertia tensor the same?

I'm looking for some help in understanding the inertia tensor (not the physics, just the math). I've seen three different definitions and I'm not sure if they mean the same thing or how they relate ...
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1answer
23 views

double root and newton method, a problem on solved exercise?

$f(x)$ in $x= \alpha$ has double roots and define in $\alpha$ neighbor. if the sequence $\{x_n\}$ for solving $f(x)=0$ calculated by newton methods the following is correct. ($a$ and $b$ is plasced ...
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0answers
25 views

Let A be an m × n matrix, and b an m × 1 vector, both with integer entries.

Let $A$ be an $m \times n$ matrix, and $b$ an $m \times 1$ vector, both with integer entries. If $Ax = b$ has a solution over $ \mathbb Z/p \mathbb Z $ for every prime $p,$ is a real solution ...
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1answer
17 views

How to find a “flag base” to an endomorphism?

I found several exercises that ask me to find a flag base for a given matrix, for example: $$ A=\left( \begin{array}{ccc} -1 & 1 & 0 \\ 2 & 2 & 4 \\ -1 & -2 & -3 \end{array} ...
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I have to show $A$ and $B$ have at least one common eigenvalue in $E$.

$A\in M_n(F)$ and $B\in M_m(F)$ be two matrices such that the minimal polynomial of $A$ divides the minimal polynomial of $B$. suppose that $E$ is the algebraic closure of $F$.I have to show $A$ and ...
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1answer
17 views

Similar matrices represent an operator relative to different bases

I need to prove the following Let $A,C$ be two similar matrices over the field $\mathbb{F}$. Define $T_A : \mathbb{F}^n_{\text{col}} \to \mathbb{F}^n_{\text{col}}$ as $T_A(x) = Ax$. ...
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2answers
25 views

Simple equation re-arrangement

I have a simple re-arrangement of an equation which I can't seem to solve, help would be much appreciated. I'm trying to re-arrange the equation: $e^{-3t}\frac{dy}{dt} - 3e^{-3t}y = C$ where $C$ ...
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0answers
20 views

Is there a fundamental meaning of kernel in “kernel function” and “kernel of linear map”

In pattern analysis kernel trick is famous, based on kernel function. On the other hand kernel of linear map is the null space. Is there a deep relation between this two "kernel" words or there is no ...
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2answers
21 views

Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix $$(A \mid I)$$ to obtain $$(I \mid C),$$ where $C$ is indeed $A^{-1}$. However, I fail ...
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3answers
37 views

Does matrix has a underlying basis?

Can I say a matrix (M) as a liner transformation and it operates on a vector? The vector must have a basis and the matrix M gave us a new vector. Now is there any basis associated with the matrix. ...
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2answers
26 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
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2answers
45 views

Show that there exists a $3 × 3$ invertible matrix $M$ with entries in $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = I_3$.

Show that there exists a $3 × 3$ invertible matrix $M$ (which is not the identity matrix) with entries in the field $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = $Identity matrix. All I could do was use ...
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1answer
19 views

Cholesky factorization exist?

Is there a theorem or a way to show that if I have a real and symmetric positive definite matrix $A$ and its Cholesky factorization is $A = LL^T$ then $B = L^TL$ is also positive definite? Or in other ...
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1answer
45 views

Linear algebra references for a deeper understanding of quantum mechanics

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?
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22 views

Spectral Radius of a Block Matrix

I have real matrix $P$ obtained from numerical solution (FEM) of a physical problem, as \begin{equation} P=P_1+P_2= \begin{bmatrix} A_{2n \times 2n}&B_{2n \times n}\\C_{n \times 2n}&D_{n ...
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0answers
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Proof of $\mathcal{R}(A^+)=\mathcal{N}(A)^\perp$ where $A^+$ is the MP pseudoinverse, is it correct?

Yesterday, I asked if that property was true : Rangespace of Moore-Penrose pseudoinverse : $\mathcal{R}(A^+)=\mathcal{N}(A)^\perp$? Someone found a demonstration, but he used SVD, and I wanted to ...
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2answers
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Proving $dim(img(g\circ f)) \leq dim(im(f))$

I want to prove: $dim(img(g\circ f)) \leq dim(im(f))$. $g$ and $f$ are linear maps. The map $f$ first maps the input $x$ into $im(f)$. $g$ take $im(f)$ into $im(g)$. Thus, this last map is ...
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1answer
20 views

How transformation of co-ordinates system relates to its vectors?

Consider a positive definite matrix. Can we consider that it has a underlying co-ordiante system? If we transform that co-ordinate system how the the vectors are transformed? Is this question even ...
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3answers
92 views

Show that if $AA^t = A^tA$, then $A=A^t$

Suppose $A$ is a matrix with non-negative real entries. If $A^tA = AA^t$, show that $A=A^t$. My proof says: $AA^t = A^tA = (AA^t)^t$. I can't seem to get to the point of $A=A^t$ Edit: What if $A$ is ...
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1answer
18 views

Meaning of co-ordinate system of Covariance matrix

Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the ...
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3answers
41 views

Find the adjoint

Choose one from he following list of inner products and then find the adjoint of: $$ \left[ \begin{array}{ c c } 1 & 2 \\ -1 & 3 \end{array} \right] $$ When your inner prod cut ...
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1answer
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How to prove the equality of two matrix expressions

I am new to linear algebra and my question maybe too simple. I have a n-by-m matrix $D$ that its columns have unit L2 norm. Let $D_a$ be a sub-matrix of $D$ composed by some columns of $D$. I need to ...
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1answer
20 views

Linear space obtained from another one factoring out the constant.

Given a three dimensional vector space $H$, I don't understand what is the two dimensional vector space obtained from $H$ by factoring out the constants. Someone can explain me that? Thanks!
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1answer
30 views

$M_n$ be space of compex n x n matrices with inner product $(A,B)= Tr A\bar{B}^t$

Let $M_n$ be space of compex n x n matrices with inner product $(A,B)= Tr A\bar{B}^t$. Find adjoint operators: i. For operator of left multiplication $L_A :X\rightarrow AX$ by matrix A for $X \in ...
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1answer
27 views

Rank of matrices, prove inequality [duplicate]

Today I'm having hard time with linear algebra problems; this is one: $\forall A,B\in M_n(\mathbb{K})$, $\mathrm{rank}(A)+\mathrm{rank}(B)\le \mathrm{rank}(AB)+n$ $M_n(\mathbb{K})$ is the space of ...
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1answer
33 views

Let P be a space of real polynomials of degree $\le 2$ with inner product [on hold]

Let $P$ be a space of real polynomials of degree $\le 2$ with inner product $\langle p,q \rangle = \int_{0}^{1}p(x)q(x)dx$ . Find volume of the set of polynomials $a+bx+cx^2$ with coefficients ...
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28 views

Is it possible to have a matrix with eigenvalues that cannot be constructed from a finite number of basic arithmetic operations, and nth roots?

For example, a characteristic polynomial $ p(\lambda) = \lambda^5 - \lambda -1 $ has the root 1.167304..., but this number cannot be written as a finite number of arithmetic operations (addition, ...
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Does there exist a steady state vector of this Markov Matrix?

Does there exist a steady state vector of Markov Matrix $$P=\begin{bmatrix} \frac{1}{2} & \frac{1}{3}\\ \frac{1}{2} & \frac{2}{3} \end{bmatrix}$$ Initially I was not sure whether to answer ...
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1answer
36 views

questions on polynomial Lagrange Interpolation of order $n$?

I ran in One Ex in my book when I‌ prepare for final exam on numerical method. how can help me how we solve such a problem? if $P(x)$ and $Q(x)$ be two polynomial Lagrange Interpolation of order $n$ ...
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1answer
32 views

Verifying linear independence for $\left\{ \sqrt{x} , \sqrt{x+1}, \sqrt{x+2} \right\}$ using Wronskian

I need to verify the linear independence for the group of functions: $$\left\{ \;f_1 = \sqrt{x} , \;f_2 = \sqrt{x+1}, \;f_3 = \sqrt{x+2} \right\}$$ using Wronskian, for $x > 0$. I wasn't told ...