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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6
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2answers
140 views

Sum of squares of maximal minors of a rectangular matrix with orthonormal rows

A matrix $A$ has $m$ rows and $n$ columns, such that $m \leq n$. We know that each row of $A$ has norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
1
vote
2answers
57 views

Question on proof about $T$-cyclic vector spaces and decomposition of minimal polynomials

Let $V$ be a vector space and $T : V \to V$ a linear transformation. We call $V$ $T$-cyclic if $V$ is generated by $\{ T^i v \}_{i \in \mathbb N}$ for some $v \in V$. For a linear transformation $T : ...
0
votes
0answers
63 views

Estimating the polynomial from only Y values

B has two polynomials of the same degree whose coefficients are defined over Zp and gives both polynomials to A. A picks a list of random X values, evaluates one of the polynomial and gives the result ...
0
votes
1answer
52 views

Question about the structure of a convex optimization problem

I am reading a tutorial about Convex Optimization and it defines the general Convex Optimization problem as: $$ minimize_x f(x)$$ where $g_1(x) \leq 0, ... , g_m(x) \leq 0$ and $Ax=b$ and $x \in ...
4
votes
1answer
43 views

Is $\operatorname{End}_K(V)$ self-opposite?

Let $V$ be a vector space over a field $K$. Is $\operatorname{End}_K(V)$ a self-opposite $K$-algebra?
2
votes
1answer
32 views

Base of Continuous functions

is there a method to determine one of the infinite bases of the vector space of continuous function over an interval [a,b] (a If the question is not "well asked" : How two functions in this space are ...
0
votes
1answer
47 views

Rank of a psd matrix and principal minors

Let $M\in\mathbf{R}^{n\times n}$ be a symmetric matrix and assume that $M$ is positive semidefinite. Let $M_i$ be the leading principal ${i\times i}$ minor and assume that $det (M_i)\neq 0$. Now if ...
0
votes
0answers
47 views

Bordered minor and rank of a matrix

Let $M\in\mathbf{R}^{n\times n}$ be a matrix. Suppose that there is a $k\times k$ minor $M_k$ of rank k. Now this reference (Algebra For Iit Jee 7.65) here states that if all the $k+1$th minors ...
2
votes
3answers
738 views

4 equations 3 unknowns

If I have 4 equations and 3 unknowns, I could solve for the 3 unknowns using the first 3. How does it ensure that the 4th equation is also satisfied? In this case, what should be the usual strategy to ...
3
votes
5answers
79 views

Linear independence of the functions $1,\cos(x),\cos(2x)$

I want to show that the functions $1,\cos(x),\cos(2x)$ are linearly independent in $C[-\pi,\pi]$. I computed the Wronskian determinat of these functions but at the points $x=0,-\pi,\pi$ the obtained ...
3
votes
4answers
111 views

Showing that this set of equations have a unique solution

I am stuck with the first part of this problem This is what I tried : I put this set of equations to a matrix and got its reduced echleon form, which is : $$\begin{array}{ccc|c} 1 & 4 & ...
1
vote
2answers
37 views

Show that $S_0$ is a supbspace of $S$

"Let $S$ denote the set of all infinite sequences of real numbers with scalar multiplication and addition defined by $a\{b_n\} = \{ab_n\}$ (where $a$ is a scalar) and $\{b_n\} + \{c_n\} = \{b_n ...
2
votes
5answers
127 views

Showing that a map, $R:\mathbb{R}^n\rightarrow\mathbb{R}^n$ can be represented by an orthogonal matrix.

Note: This is a homework question. After pages of attempts and failures, here I am. First, I will present the question then state what I have tried. The question: Let $u$ be a non-zero vector in ...
3
votes
2answers
248 views

Why do the concepts of linear algebra apply to differential equations?

A lot of the stuff we do to solve diff equations are taken word for word from linear algebra. The concept of linear independence, determinant of the Wronskian used to determine independence, adding a ...
1
vote
1answer
21 views

Angle between a vector and orthonormalized base vectors

I've been looking at this probelm for a while now and I've come up with nothing. It's a fairly simple problem from one of my old textbooks. "Let e1, e2, e3 be an orthonormalized base in the room. The ...
0
votes
1answer
62 views

Decompose the vector $\vec v = (-3,4,-5)$ parallel and perpendicular to a plane

I have the vecotr: $$\vec v = (-3,4,-5)$$ And the plane: $$\pi:\\x=1-\lambda\\y=-2\\z=\lambda -\mu$$ I need to decompose the vector $\vec v$ as the sum of a vector perpendicular to the plane and ...
0
votes
1answer
95 views

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix. $V = P_1(R)$ and $T(ax+b) = (-6a + 2b) x + (-6a + b)$ First i gave the canonical basis of ...
0
votes
1answer
35 views

Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain. ...
0
votes
2answers
77 views

Demonstrate using determinant properties that the determinant of matrix “A” is equal to, 2abc(a+b+c)^3

How can I show, using determinant properties of matrix, that: \begin{equation} \det\begin{pmatrix}(b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & ...
0
votes
1answer
40 views

Linear equation, parameter solution

I've been trying to solve a set of linear equations. But I am unsure of what method would be best suited. Yes this is a school related problem, where I have been asked to find the solution set. ...
1
vote
2answers
39 views

Solving quadratic equations in the field $F_5$

Let $y = x^2 + 2x + 2 = 0$. Solve the equation in the field $F_5$. So I used the common $b^2 - 4ac$ formula and got that $x$ is either $-1/2$ or $-3/2$ but I'm not sure if this is in the field...
1
vote
1answer
50 views

Eigenspace decomposition and nilpotent operators

In the course of reading a paper involving Markov chains, I am puzzled by a statement involving generalized eigenspaces and projections. To set the stage, let $A$ be a square matrix and denote its ...
1
vote
1answer
64 views

Linear Algebra - linear independence true/false question

Question : $$v_1,v_2,\ldots,v_n,w\in R^n$$ These are scalars $$x_1,\ldots,x_n$$ If this equation : $$x_1v_1+x_2v_2+\cdots+x_nv_n=w$$ doesn't have solution therefore $$A=\{ ...
0
votes
0answers
25 views

QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
1
vote
1answer
51 views

How to tell if two points are separated by hyperplanes?

Consider $n$ dimensional space and consider a set of hyperplanes $h_1, \dots h_k$, each one of which goes through the origin. Each hyperplane $h_i$ is defined by an $n$-dimensional vector $v_i$ which ...
1
vote
0answers
20 views

force singular value decomposition :: multiple solutions

Well I'm writing a code to solve a positioning problem. given arrival times from multiple sources I want to invert and get the receiver position. obviously I have the xyz of each receiver. so I ...
2
votes
0answers
56 views

How to prove a symmetric tensor keeps symmetric under rotation? [closed]

For example, $T_{ij}=T_{ji}$, prove $R_{il}R_{jm}T_{lm}$ is also symmetric. I know I need to prove $R_{il}R_{jm}T_{lm}=R_{jl}R_{im}T_{lm}$, and the fact that $R$ is antisymmetric might be helpful, ...
2
votes
4answers
99 views

How many solutions $X^{10} - I=0$ has in $M_2(\mathbb{R})$?

How many solutions $X^{10} - I=0$ has in $M_2(\mathbb{R})$? Where $M_2(\mathbb{R})$ denotes the set of $2 \times 2$ real matrices. I absolutely have no idea of where I should start from. $I$ and $-I$ ...
0
votes
2answers
60 views

To prove $\det (xy^t)=0$ [duplicate]

Let $x,y$ be arbitrary non-zero column vectors in $\mathbb R^n$ , then how do we prove that $\det (xy^t)=0$ ?
0
votes
1answer
38 views

Uniqueness of orthogonal projections

I'm reading a book on numerical recipes and I'm having a bit of trouble trying to prove a statement made by the authors: given $B \in \mathbb{R}^{n \times r}$ with orthonormal columns (forming an ...
1
vote
0answers
42 views

A simple test for degenerate eigenvalues of a holomorphic matrix-valued function?

Consider a symmetric $n\times n$ matrix $H(z)$ whose entries are holomorphic functions of a complex parameter $z$ and real on the real axis. It's known, from Analytic structure of the eigenvalue ...
2
votes
1answer
118 views

Bounding the multiplicative order of matrices in $\mathbb M_n(\mathbb Z)$

Let $\mathbb N$ be the set of positive integers. Prove that: $\forall n\in\mathbb{N}:\exists r\in\mathbb{N}$ (let's say $r=r(n)$ as a function of $n$) such that: If $M\in ...
0
votes
1answer
36 views

Elementary arithmetic question

2 groups of people $A$ and $B$ are trying to build a road. For the first 40 days, only one group was working at any time. At first, only group $A$ worked. They worked for an unknown amount of days, ...
1
vote
1answer
39 views

why is coherence of subspace always at least 1?

The common definition of coherence in the matrix recovery literature is as follows. Let $U$ be a subspace of $\mathbb{R}^{n}$ of dimension $r$ and let $P_U$ be the orthogonal projection onto $U$. ...
0
votes
0answers
46 views

how to get the fundamental matrix of this matrix

I have this matrix A ...
3
votes
3answers
85 views

Is the induced matrix norm continuous?

Suppose that we are dealing with positive semidefinite $n\times n$ symmetric real matrices. The induced matrix norm of $A$ is defined as $$ ...
0
votes
1answer
28 views

How to express 10/3 = 3 . by using notations like mod?

How do you express 10/3 = 3 . by using notations like mod? I need to express that k/3=something with out remainder such that 10/3=3 2/3=0 5/3=1
0
votes
0answers
18 views

What is an optimal order for integer vectors for minimization of the total distances?

I want to find an optimal order for a number of vectors (or a permutation of vectors) to minimize the sum of distances regarding to the following norm: (this norm is based on the distance on a cycle ...
0
votes
3answers
42 views

Quick question about subspaces, bases and linear independence.

Let $X$ be an $n-$dimensional vector space and $Z$ be an $(n-1)$-dimensional subspace of $X$. If $\{e_1, ..., e_{n-1}\}$ is a basis for $Z$, is it true that I can always find a vector $e_n \in X$ ...
3
votes
3answers
60 views

If the dot product between two vectors is $0$, are the two linearly independent?

If we have vectors $V$ and $W$ in $\mathbb{R^n}$ and their dot product is $0$, are the two vectors linearly independent? I can expand $V_1 \cdot V_2 = 0 \Rightarrow v_1w_1+...+v_nw_n = 0$, but I ...
0
votes
1answer
38 views

The sum of the bases of $V$ and $V^\perp$ is equal to $n$

If $V$ is a subspace of $\mathbb{R^n}$ and the size of the basis of $V$ is $l$ and the size of the basis of $V^\perp$ is $m$, then $l+m=n$. I was thinking we could have ...
0
votes
3answers
41 views

Can I find the value of $x & y$

Find x,y from N such as $x^{(2y)}=1560-x^{y}$.Is it possible to find the value of x and y only from one equation. please help me.I approached in different ways.But all my attempt went in vain.
0
votes
1answer
61 views

Finding $A$ and $b$ if we know $x$ for $Ax=b$

This is a question that appeared on a previous quiz. Nobody I know, including myself, has been able to figure it out. If $x$ is a vector in $\mathbb{R}^4$ equal to $$\left[ ...
0
votes
2answers
61 views

References for Linear Algebra needed for Differential Equations and Linear Programming

I am in need of learning the Linear Algebraic theory behind the following Applied disciplines. Could someone please recommend Linear Algebra books for: Differential Equations: Specifically learning ...
0
votes
3answers
46 views

If $V$ is a subspace of $\mathbb{R^n}$, what is $(V^\perp)^\perp$?

Is $V \subseteq (V^\perp)^\perp$? My intuition tells me that $(V^\perp)^\perp = V$, but I'm not sure if that is right. In what ways can I think about $V^\perp$ that will more easily help me understand ...
2
votes
1answer
56 views

Finding A Transforation Matrix under an Isomorphism

I am making my way through my textbook in preparation for an exam and I was wondering if anyone could assist with this problem. This problem is probably easy for most of you here, but I have been ...
0
votes
1answer
14 views

Find the point in this line such that the distance from $A$ is $\sqrt{3}$

My line: $$r: (0,2,-2) + \lambda (1,-1,2)$$ The point: $$A = (0,2,1)$$ I know that the line has equations: $$x = \lambda \\ y = 2-\lambda\\z = -2+2\lambda$$ But when I use the distance formula ...
0
votes
0answers
37 views

matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
0
votes
0answers
46 views

If I have a real symmetric matrix and one of the diagonal elements are zero, does that say anything about the eigenvalues?

In the two dimensional case, if I have a matrix $\left[\begin{array}{cc} 0 & a\\ a & b \end{array}\right]$ or $\left[\begin{array}{cc} b & a\\ a & 0 \end{array}\right]$ then I have a ...
0
votes
1answer
64 views

Linear Algebra: Vector Space, Standard Operation

I have some questions about linear algebra. 1.Determine whether the set of all third degree polynomials with standard operations is a vector space. If it is not, identify each of the vector space ...