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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2
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45 views

Prove that the eigenvectors of this matrix are a basis in $\mathbb{R}^n$

Let $A \in \mathbb{R}^{n \times n}$ and $w \in \mathbb{R}^n$. Suppose that, $w_i>0$ and $a_{i,j} = w_i / w_j$ for all $i,j=1,\dots,n$. Note that from the construction comes that ...
0
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1answer
38 views

Help to verify if this statement is correct? [on hold]

Let $S$, $T$ be two subspaces of $R^2$. Then the sum of the projections $P_S$ and $P_T$, is a projection.
0
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1answer
17 views

Prove that the columns of the similarity matrix of a diagonalization are the eigenvectors

I'm interested in eigendecomposition of a matrix. It is clear for me, that you can eigendecompose a matrix if and only if it is diagonalizable. But I don't know how to prove, that in the similarity ...
0
votes
2answers
43 views

Solve differential equation with matrix method

I have the following IVP: $$\ddot{x} + 2\dot{x} - 8x = 4$$ subject to the initial values $$x(0) = 0 \\ \dot{x}(0) = 0$$ I am asked to solve it using matrix method (I don't know if it is the correct ...
2
votes
1answer
53 views

Why do $n$ linearly independent vectors span $\mathbb{R}^{n}$?

Suppose we have $n$ linearly independent vectors $\mathbf{v}_{1}\ldots\mathbf{v}_{n}$ in $\mathbb{R}^{n}$. I know that they do span $\mathbb{R}^{n}$, because we can easily specify a non-singular map ...
0
votes
1answer
29 views

a question about linear algebra and matrix

Given a $n\times n$ matrix A,and the matrix's characteristic polynomial is $|\alpha I-A|=(\alpha-a_{1})^{r_{1}}(\alpha-a_{2})^{r_{2}}...(\alpha-a_{p})^{r_{p}}$,and $r_1+r_2+...r_p=n$. Then,as for any ...
2
votes
2answers
23 views

Basis for the vector space P2

I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial $ ax^2 + bx + c $ with the matrix $ A = \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \\ \end{bmatrix} ...
0
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0answers
10 views

Prove an equation of affine space

In affine space $\mathbb{R^n}$ there is subspace $L$ given by equation \begin{equation} c_1 x_1 + \ldots + c_n x_n + c_{n+1} = 0 \end{equation} Also, there is given affine isomorphism $f: \mathbb{R^n} ...
1
vote
2answers
49 views

Determinant is the same for $A$ and $B$?

Let us have $A=(a_{ij})$ arbitrary matrix and for $B=(b_{ij})$ matrix we have $b_{ij}=(-1)^{i+j}a_{ij}$. Prove, that $det A = det B$. I tried with some examples, and the determinant is same, but how ...
1
vote
1answer
48 views

Show that $\| u - v \|^2 = \| u - P_U(v) \|^2 + \| v - P_U(v) \|^2 $ and minimize $d(u, v)$

i) Let $\left(V, \langle\ ,\ \rangle\right)$ be an inner-product space, $v \in V$, and let $U$ be a subspace of $V$ with the orthogonal projection map $P_U$. Show that $ \| u - v \|^2 = \| u - P_U(v) ...
0
votes
1answer
77 views

Linear Algebra, multiplication of Tensor by vector by vector.

I am deriving some equations and need to know the correct mathematical notation for opening up the brackets of an equation with the following variables: tensor $A \in$ ${\mathbb R}^{l \times l \times ...
2
votes
2answers
32 views

“Simpler” geometrical description

So i was asked to find: Find the matrix that represents the linear transformation of the plane obtained by: reflecting in the line y = x, $\begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}$ then ...
-3
votes
1answer
21 views

Let $A \in {M_n}$ and $tr{A^2} = tr({A^*}A)$.why $A$ is hermitian?. [on hold]

Let $A \in {M_n}$ such as $\text{tr}(A^2) = \text{tr}({A^*}A)$. Why is $A$ hermitian?
3
votes
3answers
55 views

How to find all integer solutions for underdetermined sytsem of linear equations

I do have a system of n equations with m variables where m > n with integer coefficients. I wish to find a set of integer solutions to this system (In my case n = 2 and m = 4). Could somebody tell me ...
1
vote
2answers
36 views

How many planes are there with the desired property?

Two points $A$ and $B$ are given, for example $A(2/5/7)$ and $B(4/11/16)$. The object is to find all planes containing the points $A$ and $B$ with distance $2$ to the origin. I tried the ...
0
votes
0answers
28 views

Computations to omit from $M^TM$

I was asked to find M the the the corresponding matrix to the diagonal of S. M consists of S's eigenvectors. $$S= \begin{bmatrix} 1&2&0 \\ 2&2&1 \\ 0&1&-1 \end{bmatrix}$$ ...
1
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1answer
29 views

Three Simultaneously Diagonalizable Matrices

I have three symmetric square matrices $M$, $G$, and $S$ with the following properties: $S$: symmetric and positive semi-definite. $M$: Fully diagonal with positive entries. $G$: is a subset of ...
3
votes
4answers
85 views

Polynomial: Is there a theorem that can save my proof when $K$ doesn't include $\mathbb C$

Suppose $f(x),g(x)\in K[x]$ ($K$ a number field), let $f(x)=x^{3m}+x^{3n+1}+x^{3p+2}$, where $m,n,p\in\mathbb N$, and let $g(x)=x^2+x+1$, prove: $$g(x)\mid f(x)$$ I think this problem is not ...
2
votes
2answers
21 views

Incremental algorithm for matrix eigenvalues

I try to solve the following problem: Given a stream of symmetric matrices $A_0, A_1, ...,A_n$ such that $A_i$ is different from $A_{i-1}$ only in one place, I want to compute the eigenvalues of ...
1
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1answer
24 views

Every nonsingular $m\times m$ matrix is row equivalent to identity matrix $I_m$

Could anyone give an easy to understand proof of this fact or at least tell if the proof below is correct? We know that every matrix is row equivalent to a reduced row echelon form. An $m\times m$ ...
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votes
0answers
28 views

Linear algebra, vector subspace [on hold]

$X =\{(x_1,x_2,x_2−x_1,3x_2):x_1,x_2 \in \mathbb{R}\}$ is a subset of $\mathbb{R}^4$, $f(x_1,x_2,x_2 −x1,3x_2)=(x_1,x_1,0,3x_1)$ 1 Show that $X$ is a vector subspace of $\mathbb{R}^4$ 2 Find a basis ...
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2answers
19 views

Representing a transformation from C to C with respect to the basis 1, i

I am having trouble understanding why the transformation: $ T(z) = (3+4i)z$ from C to C can be represented by the matrix $ \begin{bmatrix} 3, -4 \\ 4, 3 \end{bmatrix}$ with respect to the basis $ ...
2
votes
1answer
37 views

calculating characteristic polynomial in $\mathbb{R}^n$

Given some hyperplane arrangement $\mathcal{A}$, we call any subset $\mathcal{B}\subseteq \mathcal{A}$ $\textit{central}$ if $$\displaystyle \bigcap_{H\in \mathcal{B}}H\neq \emptyset.$$ There is a ...
2
votes
2answers
34 views

Eigenvalues (or lack thereof) of $A$ for $A^2 = -I$

I'm just starting with Eigenvalues and Eigenvectors and it all seemed to be going fine until this question stumped me: Let A be a $2\times 2$ matrix for which $A^2=-I$. Prove that A has no real ...
2
votes
3answers
60 views

Find the inverse of a submatrix of a given matrix

I have a $A$ matrix $4 \times 4$. By delete the first row and first column of $A$, we have a matrix $B$ with sizes $3 \times 3$. Assume that I have the result of invertible A that denote $A^{-1}$ ...
2
votes
2answers
34 views

Proofs involving endomorphisms on the space of polynomials

Define endomorphisms $D$ and $E$ on the space of polynomials with rational co-efficients $ \mathbb{Q}[x] $ such that $ D(x^n)= nx^{n-1}, E(x^n) = \frac{1}{n+1}x^{n+1} $ We must show that $ DE = I $ ...
-1
votes
3answers
44 views

Linear Algebra, kernel [on hold]

Suppose that $W$ and $V$ are vector spaces, and that $f : V \mapsto W $ is a linear map. Suppose also that $u$ and $v$ are vectors in $V$ such that $f(u)=f(v)$. Show that there is a vector $w \in ...
2
votes
4answers
46 views

How to test if these are a vector space and find the basis?

I have been trying to work through these linear algebra questions in my text book for hours now, but i just cant seem to figure it out. The question is: ...
0
votes
1answer
27 views

Relation normal matrix and characteristic polynomial

If $A$ and $B$ are normal and they have the same characteristic polynomial. why $A$ and $B$ are similar?($A,B \in {M_n}(C)$)
1
vote
1answer
26 views

Proof for the necessity of conditions for a subspace

In [Axler 2015], Theorem 1.34 states that A subset $U$ of $V$ is a subspace of $V$ if and only if $U$ satisfies the following three conditions: additive identity: $0\in U$; closed ...
0
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1answer
18 views

what will be the formula for trace(T)?

Let $M_n(K)$ denote the space of all $n\times n$ matrices with entries in a field $K$. Fix a non-singular matrix $A=(A_{ij})\in M_n(K)$, and consider the linear map $T:M_n(K)\to M_n(K)$ given by: ...
0
votes
1answer
20 views

Show that there is a $\mathfrak B$ basis of a $n -$ dimmensional vector Space $V$ such that $[T]_{\mathfrak B} = A$

Let $V$ be an n-dimmensional vector space $V$ over $K$ and let $T \in \ L(V)$ such that $(T - \lambda I)^n = 0 ,$ $ \ \lambda \ \in K$ and $(T - \lambda I)^{n-1} \neq 0$. Prove that there is a basis ...
0
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0answers
12 views

splitting a system of ODEs into linear constraints and a smaller system using matrix Null Space

This problem originates from chemistry. Let us assume we want to solve a system of ODEs describing the evolution of the concentrations of the species in a chemical system with n species and k kinetic ...
0
votes
2answers
39 views

Jacobian of n linearly independent forms in n variables

Let $k$ be a field of characteristic zero and let $f_1, \ldots, f_n \in k[x_1, \ldots, x_n]_d$ be linearly independent forms of degree $d$ in $n$ variables. Is there a nice algebraic argument for ...
1
vote
1answer
19 views

Linear Transformation: Notation confusion

I'm having a hard time understanding the meaning of this notation: Let $V $ be a (possibly infinite-dimensional) vector space over a field $\mathbb F.$ Let $S: V → \mathbb F$ and let $T : V → ...
1
vote
1answer
42 views

Geometric meaning of outer product of a vector with itself

This question is related to the question in the link below: Is there a geometric meaning to the outer product of two vectors? The answer is clear, but I am wondering: If we take a outer product of a ...
0
votes
2answers
34 views

Help with basic Linear Algebra

I am teaching myself Linear Algebra and am trying to solve the following exercise from my book: Find the polynomial of degree 4 whose graph goes through the points (1, 1), (2,−1), (3,−59), (−1, ...
0
votes
1answer
28 views

find a basis of $\ker L$

I have the linear transformation: $L:\mathbb{R^4} \rightarrow \mathbb{M_2}(\mathbb{R}), \, (a,b,c,d) \mapsto \begin{pmatrix} a & c-b \\ b-c & a+d \end{pmatrix}$ I would like to find a ...
0
votes
0answers
13 views

Finding transition matrix confusion

A = {u1, u2} A' = {u1', u2'} u1 = (2 2) u2 = (4 -1) u1' = (1 3) u2' = (-1 -1) I'm trying to find the transition matrix from A' -> A and I know to do this I have to find the inverse, but I wasn't ...
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votes
1answer
25 views

I have to show nullity $B\geq 2$. [on hold]

$A$ is a non-zero, nilpotent matrix, $A\in M_n(D)$ and $n>2$, there exist $m \in N$such that $A^{2m}=0$ and $A^m\neq 0$. let $B=A^m$. because of $\operatorname{Im} B \subseteq \ker B$, I have to ...
0
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1answer
46 views

Prove there exists a unique map T

Let $V_i$ be a collection of vector spaces over field $F$ where $i=1,2,...,N$. Given the Cartesian Product $V=V_1\times V_2\times...\times V_N$ equipped with natural projections $p_i:V\to V_i$. ...
4
votes
2answers
175 views

A proof about polynomial division

Suppose $g(x)=ax+b$,$a,b\in K$,$K$ is a field, and $f(x)\in K[x]$, prove: $$g(x)|f^2(x)\Leftrightarrow g(x)|f(x)$$ The $\Leftarrow$ part is so trivial. But for the $\Rightarrow$ part I get ...
3
votes
1answer
21 views

If $ S,T \in L(V_{1},V_{2}) $, then show that $ \ker(S) \subseteq \ker(T) $ if and only if $ T = P S $ for some $ P \in L(V_{2}) $.

Let $ V_{1} $ and $ V_{2} $ be finite-dimensional vector spaces over a field $ \mathbb{K} $, and let $ S,T \in L(V_{1},V_{2}) $. Then show that $ \ker(S) \subseteq \ker(T) $ if and only if $ T = P ...
0
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1answer
32 views

Matrix transformation: not how to interpret this question

It says "By computing the eigenvalues and eigenvectors of the matrices, give a geometrical description of the linear transformation associated with the matrices" And then i was given 4 2/2 matatrice ...
3
votes
1answer
21 views

If $A$ is normal matrix and $A$ has distinct eigenvalue and $AB=BA$.why $B$ is normal.

If $A$ is normal matrix and $A$ has distinct eigenvalue and $AB=BA$.why $B$ is normal?($A,B \in {M_n}$)
1
vote
1answer
30 views

What are the differences between the 3 “versions” of “Finite-Dimensional Vector Spaces” by P.R. Halmos?

There are 3 versions on amazon: the 1st one is published at 1993 Aug, by Springer, noted "1st ed. 1958. Corr. 2nd printing 1993 edition (August 20, 1993)"; the 2nd one is published 2014 April by ...
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0answers
29 views

What is meant by an eigenvalue of 2 matrices?

In looking for a way to compare covariance matrices, I came across a paper that formulates a metric using what appears to be a joint eigenvalue. I'm not familiar with this idea. Thus we propose ...
0
votes
1answer
16 views

Are approximate least square intersections unique?

I seem to be getting a different approximate intersections for the same three lines by multiplying one of the line equations (so that the equation still defines the same line but has different numbers ...
0
votes
0answers
86 views

Understanding Eigenvector

We have a matrix $A$ of size $N \times M$, where $N\le M$. Consider a vector $V$ of length $N$. Now I take product of $AV$ to get a vector $W$ of length $M$. Here I have projected the original ...
1
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0answers
17 views

How many distinct possible forms for its Jordan canonical matrix are there? 4x4 non-diagonalizable matrix with two unique eigenvalues

I know the sum of $A_m$ equals $4$ as $\dim(A) = 4$ and sum of $G_m$ can't equal $4$ as $A$ is non-diagonalizable. After I write down all the cases, what should I do?