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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25 views

Find spectrum for 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\\ ...
2
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0answers
34 views

Find x,y & z (xyz+xyz=zyx)

I saw this problem the other day at work and found it pretty interesting: xyz + xyz = zyx Find x, y, z and the base(s) which this is true. Note that x,y,z are ...
0
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0answers
28 views

How to find out if this vector system of functions is linearly independent?

For example, i have these functions in the vector space $\mathbb R^{\mathbb R}$: $x^2-x+3$, $2x^2+x$, $2x-4$ And I have to determine if they are linearly independent, how should I solve this ...
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0answers
7 views

If $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are in $d$-general position, then they are in $1$-general position.

Let $\mathcal{L}_{d}^{n}$ be the $\binom{d+n}{n}-1$ dimensional projective space of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ and $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$. We denote by ...
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0answers
8 views

Bind Variable and Free Variable, A Questions and one Example?

I see a Local Contest Question as : for statement $ \forall x [ \exists y ( x<y+z) \to \exists z (x < y+z)] $ two following axiom is True: I) $ y, z$ is free and $x$ is bounded variables. ...
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2answers
19 views

Find matrix representation of transformation

Given two lines $l_1:y=x-3$ and $l_2:x=1$ find matrix representation of transformation $f$(in standard base) which switch lines each others and find all invariant lines of $f$ My attpempt is to ...
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2answers
27 views

Prove linear transformation

I'm working on linear transformation trying to answer : Let $E$ and $F$ be two vector spaces on $\mathbb{K}$ and $L:E \rightarrow F$ a function. The graph of $L$ is $\mathbb{G}(L)=\{(x,y) \ \in \ ...
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0answers
12 views

Show convergence of Power method

Given a symmetric positive definite matrix $A_0 \in R^{n \text{x} n}$ with Cholesky decomposition $A_0 = LL^T$. How can I show that $A_k$ converges to $diag(\lambda_1, ..., \lambda_n)$ where $A_k$ is ...
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1answer
13 views

proof involving field

let $A$ and $B$ be elements of a field, and suppose that $AB=0$. Prove that at least one of $A$ and $B$ must be equal to $0$. Here is my answer: $AB=0$, $AB=A.0$, $AB-A.0=0$, $A(B-0)=0$, hence either ...
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4answers
23 views

Basis of a $2\times2$ matrix

How would I find the basis for an arbitrary matrix W such that: $$ W =\left\{ \begin{pmatrix} a & b \\ c & a +b +c\end{pmatrix} \ \big| \ \ a ,b ,c \in \mathbb{R} \right\} $$
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1answer
41 views

Is $\text{rank} (AA^*)=\text{rank}(A)$ for all nonsquare matrices?

If $A$ is a $m\times n$ type matrix with $m\geq n$ then $$ rank (A^*A)=rank (A). $$ Is maybe also true in general that $$ rank (A^*A)=rank (A) ? $$ Thanks Edit. My question is different from the ...
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0answers
21 views

How do I fit this piece of code on one line in Latex [migrated]

I am trying to have three matrices on one line as i) A=, ii) B= and ii) C=. I tried \nopagebreak, \noindent just after item. Instead I always get the Roman numeral on one line, a comma on the next ...
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0answers
10 views

Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...
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votes
2answers
24 views

Proove L is a linear transformation

I'm working on linear transformation and trying to answer : Let E and F be two vector-spaces on $\mathbb{C}$ and $L:E \rightarrow F$ an application such as : $\forall u,v \in \ E, L(u+v)=L(u)+L(v) $ ...
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0answers
16 views

Equating eigenvalues of Hermitian matrix and correlating symmetric/antisymmetric matricies

I have a matrix $AH$ which is created by adding $AS$ and $i*AA$, which are the symmetric and antisymmetric components of the matrix $A$ So $AS=(A+A')/2$ $AA=(A-A')/2$ $AH=AS+i*AA$ AH has ...
0
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1answer
12 views

matrix sampling and its rank preservation

Assuming matrix $X\in R^{m\times n}$ is row orthogonal of rank $m$. Then, if I construct a new matrix $Y\in R^{m\times t}$, whose columns are directly sampled from $X$ with or without replacement ...
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2answers
37 views

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
votes
3answers
59 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
28 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
50 views

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
26 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
12 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
23 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
19 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
16 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
11 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 ...
1
vote
1answer
24 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
16 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
22 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
20 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
18 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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votes
2answers
72 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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3answers
38 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
27 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
16 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
22 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
23 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
24 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}) = ...
0
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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
41 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
27 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\\ ...
1
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0answers
24 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 ...
0
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0answers
14 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 ...
0
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1answer
33 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
votes
0answers
46 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
19 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 ...
0
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2answers
43 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 ...
5
votes
4answers
156 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 & ...