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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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 ...
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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 ...
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14 views

Example of a special kind of infinite dimensional vector space and a linear map on it

Give example of an infinite dimensional vector-space $V$ and a linear transform $T$ on $V$ such that $T \circ S=S\circ T , \forall S \in \mathscr L(V) $ , but $V$ has a non-zero vector which is not ...
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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 : As you can see it doesnt seem to ...
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2answers
15 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 ...
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1answer
39 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 ...
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3answers
143 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 ...
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13 views

Lagrange Multiplier Method

How do I show the following? Show by Lagrange multiplier method that the maximum value of $\frac{d\phi}{ds}$ is $|\nabla\phi|.$ So $s$ is the distance and $\frac{d\phi}{ds}=\nabla\phi\cdot ...
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1answer
11 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 ...
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20 views

$range(M^tM)=range(M^t)$

Let $M$ be a $n\times n$ matrix with real coefficients and let $M^t$ be the transpose matrix, i.e., $(M^t)_{ij}=M_{ji}$ . Prove that $range(M^tM)=range(M^t)$. I really don't know how to even start ...
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26 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 ...
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12 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 ...
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1answer
22 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. ...
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1answer
17 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. ...
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2answers
29 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...
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1answer
18 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 ...
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1answer
43 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=\{ ...
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11 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 ...
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1answer
32 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 ...
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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 ...
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30 views

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

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, ...
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1answer
16 views

Rigid motion question [on hold]

Does anyone know how to show the following? Let $\mathbb{R}^n$ be n-dimensional Euclidean space with the inner product $\langle , \rangle$ given by the dot product. A transformation $T:\mathbb{R}^n ...
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4answers
85 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 $-I$ ...
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2answers
50 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 $\det (xy^t)=0$ ?
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Proff of the following theorem with regards to Linear dependence [on hold]

Can someone please prove the following theorem: If S is linearly independent if and only if S has a finite linearly dependent subset
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1answer
31 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 ...
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0answers
25 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 ...
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1answer
95 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 ...
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1answer
25 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, ...
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1answer
15 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 $U$. ...
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2answers
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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 $$ ...
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1answer
22 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
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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 ...
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3answers
22 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 $e_n \in X$ ...
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3answers
37 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 ...
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1answer
33 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 ...
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3answers
33 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.
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1answer
53 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[ ...
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2answers
25 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 ...
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3answers
40 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 ...
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1answer
28 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 ...
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1answer
11 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 ...
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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 ...
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23 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 ...
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1answer
21 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 ...
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3answers
57 views

ker$(A^n) =$ ker$(A^m), \forall m > n$

If $A$ is a square matrix and ker$(A^n) =$ ker$(A^{n+1})$, then ker$(A^n) =$ ker$(A^m), \forall m > n$. I'm trying to prove that this is correct, but I'm having trouble figuring out what the ...
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2answers
16 views

Are subspaces $X$ and $Y$ closed under addition?

If $X$ and $Y$ are each a subspace of $\mathbb{R^n}$, is $X+Y$ also a subspace of $\mathbb{R^n}$, where $X+Y$ is the set of all vectors $x+y$ such that $x\in X$ and $y\in Y$? I've already ...
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22 views

Solving a simple Distance Geometry problem

I'm trying to solve the following problem: Given the absolute positions of four points in 3D space, and the distances from these four points to a fifth point, find the position of the fifth point. ...
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1answer
29 views

dimension of direct products

Suppose $\{V_i\}_{i\in I}$ is a family of $k$ vector spaces. Is it possible to calculate $\dim\oplus_{i\in I} V_i$ and $\dim\prod_{i\in I}V_i$?