Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Reconstruct a vector with a known vector and residual

I observe $\vec y \in \mathcal R^n$ and know $\vec x$. I assume that $\vec y$ mostly consists of $\vec x$, with some added residual $\vec r$. This gives me the problem $\vec y = a\vec x + \vec r$, ...
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0answers
18 views

prove or disprove that a particular invertible matrix is also orthogonal

is it true that, if for some $2n \times 2n$ matrices $O^t=O^{-1}$ and $$J_0= \begin{bmatrix} \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & ...
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2answers
24 views

Finding Cases Of Inequality Between Null Space And Solution Set

$H=(h\in R^m; Ah=0)$ $L=(l \in R^m; Al=b)$ Find a matrix $A^{n*m}$ so: $|L|=0 < |H|=1$ $|L|=0 < |H|=\infty$ $|L|=0 < |H|=7$ As for 1. \begin{pmatrix} 1 & 2 \\ 0 & 0 ...
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2answers
71 views

Proving Equality About The Null Space

let there be a matrix $A^{n*m}$ that $Ax=b$ the solution set of the homogeneous system $H=(h\in F^m; Ah=0)$ the solution set of the non-homogeneous system $L=(l \in F^m; Al=b)$ Prove: if $l_0\in ...
4
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1answer
41 views

Does $(x,f(x),\cdots,f^p(x))$ is linearly dependent over $E$ implies $(id, f, …, f ^ p)$ is linearly dependent over $\mathcal{L}(E)$?

Here is the original (classic I think) problem I had encored: if $(x,f(x))$ is a linearly dependent family of $E$ (a vector space) for all $x\in E$, then the family $(id,f)$ is linearly dependentt ...
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35 views

Matrices - removing columns does not affect row equivalence [on hold]

Could anyone explain why? For example, I can imagine two row equivalent matrices. Row vectors of row equivalent matrices are linearly dependent. What if I can add a new column to the first matrix, and ...
2
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2answers
24 views

Forming a new matrix by adding the same number to any row or column

Say that two $m\times n$ matrices, where $m,n\ge 2$, are related if one can be obtained from the other after a finite number of steps, where at each step we add any real number to all elements of any ...
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28 views

Matrix that needs to be reduced to reduced row echelon form

What does the first column mean? Do I move the first column to the last column?
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1answer
34 views

Finding general solution to diff equation [on hold]

$3y'+2y = 4y^{1/8}$ How do you solve this differential equation?
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2answers
46 views

Find a vector parallel to the intersection of the planes $2x-3y+5z=2$ and $4x+y-3z=7$

The solution is $(4,26,14)$. I know how to find the intersection of the planes, but not a parallel vector.
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1answer
11 views

In what case for a matrices p1 and p2(they are with the same size) on R, we can find invertible matrices A and B such that p1=Ap2B?

In what case for a matrices p1 and p2 on R, we can find invertible matrices A and B such that p1=Ap2B?Is that always true?
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30 views

Prove that if $m<n$ then $S$ does not generate $V$

Let $V$ be a vector space over a field $K$ such that $dim V=n$ and let $S\subseteq V$ such that $|S|=m$. Prove that if $m<n$ then $S$ does not generate $V$ Let $S=${$s_1,...,s_m$}. Suppose that ...
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26 views

Positive Definite Matrices Properties that I'm trying to prove right/wrong:

Is the product of two: (a) positive definite matrices positive definite? (b) symmetric positive definite matrices positive definite? (c) symmetric positive definite matrices symmetric positive ...
1
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1answer
35 views

Endomorphism ring of finite-dimensional representation

If $G$ is any group and $V$ is a finite-dimensional representation of $G$, then we can form the endomorphism ring $E = End_G(V)$. Suppose that $V$ is indecomposable, i.e. not a direct sum of ...
1
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1answer
22 views

definition of linearly dependent set

I know that this is a silly question to ask but I would really appreciate if you can answer me. Let $V$ be a vector space and $S=\{v_1,\ldots,v_n\}$ a finite subset of $V$. $S$ is linearly dependent ...
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0answers
19 views

Some theorem of Newton (discrete Taylor expansion)

Let $\Delta$ be the forward difference operator, $\Delta f(x)=f(x+1)-f(x)$. Is there an elegant way to prove that for every $f\in\mathbf{Q}[x]$ (of degree $n$, say) the equality $$f(x)=\sum_{k=0}^n ...
2
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1answer
26 views

Obtain a special solution for a differential equation

If I have a differential equation $y" + 4y' + 4y = 2e^{-x} + xe^{-x}$ and $f(x)$ is a special solution to this equation, how can I determine $f(x)$ if I know f(0)=0 and $f'(0)=0$ ?
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1answer
14 views

Spectral Decomposition of Function-of-Normal-Operator

In Arveson's book A Short Course on Spectral Theory, on page 64 (section on spectral measures) the author mentions the usual spectral decomposition of a normal operator $N$ as $$N=\sum_{\lambda \in ...
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0answers
56 views

Reducing a linear algebra expression to quadratic form

I am trying to solve the following exercise for my Machine Learning course. Expand this expression so that there are only quadratic terms: $(\mathbf{x} - \mathbf{\mu})^T \mathbf{\Sigma}^{-1} ...
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17 views

Quadratic forms for integral [closed]

I can not solve ex.2 number 6 at (b) if i start from given then i arive to q(p) not q1(p) But in the question q1(p) It is imposaible please any person solve this ex. In image
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1answer
19 views

Tools to compress a finite list as a function

Can someone show me some tool to a lossless compression in an algorithm of a finite list of rational numbers? By example this list A=(0,1,3,2,-1,-2,0), there is a way to construct an algorithm or ...
3
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2answers
58 views

$\dim (D-P)=\dim (D)-1$

I'm trying to prove this question: Let $D$ be a divisor in $F|K$ such that $\dim (D)\gt 0$ and $0 \neq f\in \mathscr L(D)$. Thus $f\notin \mathscr L(D-P)$ for almost all $P$. Then show that ...
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1answer
30 views

Perturbation of Determinant

Suppose we have a linear equation with parameter $0 <\lambda <1$ as $\left(\begin{array}{ccc} 3-\lambda & -1 & -1\\ -1 & 1-\lambda & 0\\ -1 & 0 & 1-\lambda ...
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1answer
12 views

Invariant spaces of two linear transformations

Let there be $S:V \to V$ and $T:V \to V$ linear transformations ($dimV=n$), and let there be $W$ a invariant subspace of both $S$ and $T$. Is it true that $TS=ST$? Can it be proven?
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Preimage of an affine subspace

Let $V$ and $W$ be vector spaces. Prove: If $B$ is an affine subspace of $W$, and if $T \in \text{Hom}\,(V,W)$, then $T^{-1}[B]$ is either empty or an affine subspace of $V$. Would someone give me ...
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1answer
37 views

The level set of a smooth function

Let $f$ be a smooth function on a manifold $M$. Fix a point $p\in M$ and let $df\in T^\ast_pM$ be the differential of $f$ at $p$. I read that the subspace of $T_pM$ of vectors $X$ such that $df(X)=0$ ...
2
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2answers
60 views

The minimum of $x^2+y^2$ under the constraints $x+y=a$ and $xy=a+3$

I solved the following problem: If $x,y,a \in \mathbb{R}$ such that $x+y=a$ and $xy=a+3$, find the minimum of $x^2+y^2$ Here is my solution. $x^2+y^2=(x+y)^2 -2xy= a^2-2a-6$. The minimum value is ...
1
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1answer
21 views

How to prove that a given set of functions forms a linear subspace of a larger space?

I need to prove that these $S$s are Linear subspaces of $V$. I tried: 1- I was thinking: I can choose any scalar number who belongs to real numbers, so, if I to multiply $p(x)$ by a scalar, I have ...
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0answers
21 views

Basic Affine Combinations Question

First off let me say that I am NOT looking for the answer to just be given to me, I just need help figuring out where to start, as all the explanations in the test are a little too abstract for me. My ...
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2answers
33 views

Vector space and vector subspace 1111 [closed]

I have to know if this is or not a vector space!!!!!!! How can I solve this question: V=P(R³) e S={p e P(R³): the coefficient in the term x is zero}? P(R³), generated for the polynomials from R³.
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1answer
53 views

Determine all the integer solutions to $23x + 39y = 2$

I would like to calculate all the solutions to this equation using Euclides' algorithm and linear combination after finding the GCD. I suppose it's easy, but I'm a beginner. $23x + 39y = 2$
4
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1answer
33 views

Normal transformation

Let $V$ be a finite-dimensional vector space over $\Bbb C$ and $T:V\to V$ be a linear transformation. Assume that every eigenvector of $T$ is also an eigenvector of $T^*$. I need to prove that ...
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1answer
44 views

Prove with combinatorial arguments this equation [duplicate]

Prove with combinatorial arguments, that, $\forall n \in \mathbb{N}$. $$\sum_{k=0}^n (-1)^k {n \choose k} =0$$
2
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2answers
38 views

Show formula of inverse matrix

Let p(Y)=(-Y)^n + a_(n-1)*Y^(n-1) + ... + a_0 be the characteristic polynomial of matrix A. Show that A is invertible, if and only if a_0 isn't zero and that inverse of A is A^(-1)=q(A), where q ...
3
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2answers
46 views

example of complex structure with negative determinant

is it possible to find a matrix $J_1 \in GL(4,\mathbb R)$ such that $\det J_1=-1 $ and $J_1^2=-\operatorname{id}$ ? if it is, how can we prove that every matrix $M \in GL(4,\mathbb R)$ such that ...
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0answers
13 views

Simplify recursion function based on a matrix, real-world usecase

I have an auction running, and I'm trying to calculate the expected amount of first, second etc. places to be taken by a particular bid. To achieve that, based on historical data I make a following ...
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0answers
18 views

Irreducible matrices and connected graphs

The adjacency matrix of a simple undirected graph is irreducible if and only if the graph is connected. Here my questions : Is there an efficient method to check whether a matrix is irreducible (I ...
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0answers
26 views

How to construct a function to map coefficients?

Surely this question is known by many people but I lack of enough maths knowledge so I prefer ask here. I have a triangular matrix that represent coefficients, all of them are rational numbers ...
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3answers
29 views

Concerning the point $(7,a)$ on the line containing $(0,0)$ and $(4,2)$

I have recently been studying to take the GRE's and while working through the math section I find a lot of problems similar to this: Now I know it is supposed to be assumed that point $O$ is marked ...
1
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1answer
39 views

Find the values for which the matrix A is positive definite.

I would like to know how I can solve this exercise: Find the values of $\lambda$ so that the matrix is positive definite $$A= \left[ {\begin{array}{cc} 2 & \lambda & -1 \\ ...
0
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1answer
33 views

How to prove the equivalence of 2 affine spaces given that one is the subset of the other one?

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
5
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3answers
253 views

How to encode matrices uniquely

Given a square matrix $A=[a_{ij}]_{n \times n}$, an operation $swap(A, i, j)$ is defined to swap row $i$ and $j$ of $A$ and do the same thing with the corresponding columns. For example, in the ...
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2answers
36 views

A criteria for a subalgebra of M(n,C) being M(n,C)

Suppose $S$ is a subalgebra of the matrix algebra $M_n(\mathbb{C})$. If for any vector $v$ and $w$ in $\mathbb{C}$, there always exists a matrix $A$ in $S$, depending on $v$ and $w$ of course, which ...
3
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1answer
35 views

What should be the characteristic polynomial for $A^{-1}$ and adj$A$ if the characteristic polynomial of $A$ be given?

Let the characteristic polynomial of $A$ be $\psi_A(x):=p(x)$. If $A$ be non-singular, then find that the characteristic polynomial of $A^{-1}$ and adj$(A)$. My attempt: We have \begin{align*} ...
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4answers
44 views

Straight lines - equation of line

Question: A line $4x + y = 1$ through the point A(2, -7) meets the line BC whose equation is $3x - 4y + 1 = 0$ at point B. Find the equation to the line AC, so that AB = AC. I can't even understand ...
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1answer
21 views

properties of largest eignvalue of product of two matrices

I'm searching for the proof of this lemma it's about largest eignvalue of product of two matrices. one of them is positive definete and the other one is symmetric. B is symmetric matrix, A is Positive ...
5
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1answer
49 views

Prove that $\sin(x+a), \sin(x+b),\sin(x+c), \hspace{5pt} a,b,c \in \mathbb{R}$ are linearly dependent

I just want an answer verification (or not). We have that: $\begin{array}[t]{l} \sin(x+a)=\cos a\cdot \sin x + \sin a \cdot \cos x\\ \sin(x+b)=\cos b\cdot \sin x + \sin b \cdot \cos x\\ ...
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1answer
24 views

Finding the solutions of a system of linear equations with an unknown k. Please help me

Consider the system of linear equations $x + 2y = 2$ $-x + (k-3)y + 3z= -2$ $2x + (k + 3)y + (k+3)z= 5$ where $k$ for all real numbers is a constant. Determine the values for $k$ for which the ...
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0answers
23 views

hoffman-and-kunze[Chapter 6][Problem 6.8.4] [closed]

Let $T$ be a linear operator on a finite-dimensional space $V$ with characteristic polynomial: $$f=(x-c_1)^{d_1}\dots(x-c_k)^{d_k}$$ and minimal polynomial $$p=(x-c_1)^{r_1}\dots(x-c_k)^{r_k}$$ Let ...
2
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2answers
44 views

Find the linear transformation from kernel and range

Find the linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4 $ with $\ker T = [(1,0,1,0),(-1,0,0,1)] $ $\operatorname{Range}T = [(1,-1,0,2),(0,1,-1,0)] $ So if $v \in \ker T$, then ...