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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Prove that the output of the function equals the determinant

Let $δ$ : $M_{2×2}$($F$) $→$ $F$ be a function with the following three properties. ($i$) $δ$ is a linear function of each row of the matrix when the other row is held fixed. ($ii$) If the two rows ...
2
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3answers
53 views

Logic behind finding a $ (2 \times 2) $-matrix $ A $ such that $ A^{2} = - \mathsf{I} $.

I know the following matrix "$A$" results in the negative identity matrix when you take $A*A$ (same for $B*B$, where $B=-A$): $$A=\begin{pmatrix}0 & -1\cr 1 & 0\end{pmatrix}$$ However, I am ...
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1answer
26 views

Gauss Method to show [on hold]

Could you please give me the way to solve this problem Using Gauss method to show if $x ≠ y + 1$ then $$ \sum_{i=0}^n (x-y)^i = \frac{(x-y)^{n+1}-1}{x-y-1}. $$
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2answers
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how to convert log(x) into linear form? [on hold]

I have simple function which is non-linear like log(x) I want to convert it into linear function. Anyone could help out? Thanks
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1answer
25 views

Reflection matrix in $ \mathbb{R}^{3} $.

I need help in understanding how they got the transformation matrix $ Q_{L} $ from Theorem 2 and $ P_{M} $ at the bottom of the page. They skipped some steps and I find it confusing. Any help would ...
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0answers
8 views

Can independence of a system and a vector be establish if there is only cross-indepedence?

Say that I have the following linear system: $$[A a'] \begin{bmatrix} x \\ x' \\ \end{bmatrix} =Ax + a'x' $$ I want to know when this system is zero if and only if ...
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1answer
21 views

Eigenfunctions and Eigenvalues of a Linear Operator

For a math project on Schroedinger's equation I and a partner are working on we need to find eigenfunctions and eigenvalues that satisfy $L\phi_n = \lambda_n\phi_n$, where $L$ is defined as $L\psi = ...
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2answers
18 views

Algebra verbal find the amount of sold items

Hey I am having an exam tomorrow, so I looked up at some verbal algebra questions, and found one that I could not solve, because I don't really understand how would I do this. The question is like ...
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0answers
15 views

About lower bounds on the size of irreducible representations of subgroups of symmetric groups.

Is there a subgroup $G_n$ of $S_n$ (one $G_n$ for each $S_n$) increasing in size such that their permutation representations are such that the smallest non-trivial irreducible size in them is ...
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0answers
6 views

Characterise the set of inner products which are preserved by a given automorphism?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. (You can ...
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1answer
23 views

Finding eigenvalue and eigenvectors of a matrix containing an imaginary number

How do you solve for the eigenvalues given the matrix? \begin{matrix} i & -2 \\ 1 & 0 \\ \end{matrix} I know how to get the characteristic polynomial Ca(X); X^2 - ...
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2answers
15 views

Approximating length of a curved line based on Begining and End points of line

I have two points, a known distance apart. At each of these points I have a sensor that gives me flow speed and direction. I originally assumed the flow path between the first point and second point ...
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0answers
20 views

Change of eigenvectors by change of coulmn vectors.

This question is an extension to the question in the link: Change the matrix by multiplying one column by a number. It is understood now that if we change a positive definite matrix A to B by ...
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0answers
19 views

How to find spectral radius of ${0,1}$ and ${0,1,-1}$ matrices?

[this is kind of a continuation of this question ] It seems that the following is true, Among $n=3$ dimension symmetric matrices over $\{0,1\}$ which have $d=7$ ones the maximum spectral radius is ...
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1answer
16 views

linear algebra characteristic values [on hold]

Let $T$ be the linear operator on $\mathbb{R}^4$ which is represented in the standard ordered basis by the matrix $$ \left( \begin{matrix} 0 & 0 & 0 & 0 \\ a & 0 ...
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2answers
71 views

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
36 views

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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0answers
8 views

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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1answer
26 views

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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0answers
15 views

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
18 views

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
22 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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1answer
21 views

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
21 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
25 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
33 views

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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0answers
22 views

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
23 views

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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0answers
13 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 ...
2
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0answers
33 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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0answers
17 views

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
41 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 ...
4
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3answers
137 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 & ...
2
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0answers
22 views

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'm trying to figure out how to convert between the wedge product and tensor product definitions. ...
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1answer
25 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 ...
2
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0answers
28 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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0answers
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I have to show $A$ and $B$ have at least one common eigenvalue in $E$. [on hold]

$A\in M_n(F)$ 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
18 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
22 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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24 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
30 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
22 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
56 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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25 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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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 ...