Tagged Questions

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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0
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3answers
370 views

Finding a unique representation as a linear combination

ok, another problem suggested by my prof. the vectors $u_1 = (1,1,1,1)$, $u_2 = (0,1,1,1)$, $u_3 = (0,0,1,1)$, $u_4 = (0,0,0,1)$, are a basis for $F^4$. Find a unique representation of an arbitrary ...
6
votes
2answers
540 views

$A$ is some fixed matrix. Let $U(B)=AB-BA$. If $A$ is diagonalizable then so is $U$?

This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze. Here is the question. Let $V$ be the space of $n\times n$ matrices over a ...
0
votes
1answer
106 views

Could someone help me to prove that this symmetric matrix is definite positive?

Let $a_{ij}\in\mathbb{R}$ for all $i,j\in\{1,...,n\}$ and $m\in\mathbb{N}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
3
votes
2answers
9k views

How to check if a set of vectors is a basis

OK, I am having a real problem with this and I am desperate. I have a set of vectors. {(1,0.-1), (2,5,1), (0,-4,3)} How do I check is this is a basis for $R^3?$ My text says a basis B for a vector ...
2
votes
0answers
163 views

This matrix is an attractor?

I'm trying to find for which values of $\gamma$ the matrix A is an attractor: $$ A=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ -1 & 0 & \gamma ...
1
vote
0answers
33 views

the det of identity replacing column

Let $A\in{\mathbb{R}}^{n\times n}$, $b\in\mathbb{R}^{n}$, and $x\in\mathbb{R}^{n}$ be given, where $A$ is nonsingular and $Ax=b$ holds. Let $X_{j}$ be the matrix obtained from the $n\times n$ identity ...
4
votes
1answer
97 views

$\delta$ Notation in linear algebra

In this equation below, what is $\delta_{l,q}$ denoting? Is $\delta$ a standard notation, or anything to do with all one's or the basis matrix etc? $$A_{ij}=\delta_{l,q}\left(\sum_{h=1}^n B_{l,h} + ...
5
votes
4answers
285 views

Every element of $U + V + W$ can be expressed uniquely in the form $u + v + w$

Suppose that $U$, $V$ and $W$ are subspaces of some given vector space. With the obvious definition of $U + V + W$, show that every element of $U + V + W$ can be expressed uniquely in the form $u + v ...
1
vote
1answer
25 views

Given a specefic set $ A$ we need to find $A^\perp$

Suppose we have a set of functions which are an element of $L^2[0,1]$ where if we let f(x) be the function equal to 0 from $0<x<1/2$. If this set A is a subset of the Hilbert space $L^2[0,1]$ ...
2
votes
0answers
102 views

Interpret the Parallelogram Law

I'm starting to teach myself linear algebra from a book but I don't understand the following question. Interpret $||A+B||^2+||A-B||^2 = 2||A||^2 +2||B||^2$ as a "parallelogram law." Any suggestions ...
5
votes
0answers
105 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
1
vote
2answers
203 views

Orthonormalize the set of functions {1,x}

I'm having trouble while doing this exercise, it says: In the vector space of the continuous functions in [0,1] with the inner product : $$\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$$ a) ...
2
votes
1answer
146 views

Intersection of orthonormal basis of two subspaces

I want to ask how can I form a subspace which is an intersection of the orthonormal basis of two another subspaces and find its dimension in Matlab? Thank you in advance, Maya
2
votes
2answers
158 views

Finding eigenvectors of similar matrices

Suppose I have two matrices $B$ and $ABA^{-1}$ where $A = [v_{1}\, v_{2}\, v_{3}]$ with $v_{i}$ column vectors such that $\{v_{1}, v_{2}, v_{3}\}$ form an orthonormal basis of $\mathbb{R}^{3}$ (not ...
1
vote
0answers
47 views

Edge in a convex polytope

I want to show that a convex polytope $A$ that is an intersection of half-spaces contains an edge if $ A=\{x \in \mathbb{R}^n|Ax=0 \wedge x \ge 0\}$, where x greater equal 0 means, that all components ...
2
votes
1answer
281 views

Can we 'form' infinitely many subspaces out of finite dimensional vector space?

Let $V$ be a vector space over $\mathbb{R}$ of dimension $n$, and let $U$ be a subspace of dimension $m$, where $m < n$. Show that if $m = n − 1$ then there are only two subspaces of $V$ that ...
2
votes
2answers
45 views

How to calculate these matrices? - explanation of the procedure

Can you please help me solve this problem? I have got these matrices $A=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 2 \\ 3 & 0 & 1 \end{array}\right) $, ...
0
votes
1answer
146 views

Symmetric positive definite with respect to an inner product

Let $A$ be a SPD(symmetric positive-definite) real $n\times n$ matrix. let $B=LL^T$ be also SPD. Let $(,)_B$ be an inner product given by $(x,y)_B=x \cdot By=y^T Bx$. Then ...
1
vote
1answer
69 views

Does nullity at one point implies nullity everywhere?

Consider the following definition of the derivation. Now, consider a derivation $\delta_{p} :C^{r}(R^{n})\rightarrow R$ , i.e. it is defined on r-times differentiable functions defined on the ...
-1
votes
2answers
75 views

Why is $a^2 + b^2 = c^2$ (Linear algebra)

This definition of the norm holds for the dot product in Euclidean space. It does not feel like a axiom, because if we defined distance differently we clearly run into problems. It does not hold for ...
0
votes
1answer
36 views

Relationship between adjoing matrix and inverse function

I am struggling with the following excercise: Let A be a matrix, then we have for every subspace $U$ that: $A^*(U ^\perp)=(A^{-1}(U))^\perp$ I do not even know where to start to solve this ...
0
votes
1answer
220 views

Relationship between an inhomogeneous Poisson process and Markov chain

What type of Markov process relates to an inhomogeneous Poisson process? A homogeneous Poisson process-- one where the rate, $\lambda$, is constant-- is a pure birth continuous time Markov chain ...
0
votes
1answer
49 views

Verifying spectral norm

I was wondering how one could verify the relation that $||A||_2 = \sqrt{\rho(A^HA)}$ for matrices. I mean I have seen this so often, but never found a proof of it. Is there a smart way to do this ...
5
votes
2answers
501 views

How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?

In our lecture notes, there's the following example problem. Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 ...
3
votes
0answers
80 views

The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: ...
2
votes
1answer
108 views

Matrix of quadratic form has to be symmetric?

On Wikipedia it is stated that any $n\times n$ real symmetric matrix A determines a quadratic form. But isn't $ax^2 + bxy + cxy + dy^2$, the quadratic form given by $v^T A v$ with $A=\begin{bmatrix}a ...
1
vote
1answer
252 views

Linear transformation?Rotation question in linear algebra

Given the vector $x=[1, 3, -7]^T$ in the basis $[1, 0, 0]^T$, $[0, 1, 0]^T$, and $[0, 0, 1]^T$ (Cartesian coordinates) perform the following operations: Rotate 45 degrees about the x-axis then rotate ...
4
votes
1answer
146 views

Examples of how to calculate $e^A$

I'm trying to learn the process to discover $e^A$ For example, if $A$ is diagonalizable is easy: $$A =\begin{pmatrix} 5 & -6 \\ 3 & -4 \\ \end{pmatrix}$$ Then we ...
2
votes
2answers
81 views

Is there a quick way to compute the matrix whose column space is the basis of the null space of another matrix?

Is there a quick way to compute the matrix whose column space is the null space of another matrix? I can do this by hand, but if I wanted a computer to do it, is there a quick, efficient way for me ...
3
votes
2answers
57 views

Find a subspace $W$ such that … - am I right?

Let $U$ be a subspace of $\mathbb R^4$ spanned by $v_1=(1,-1,1,2)$ and $v_2=(3,1,2,1)$. Find a subspace $W$ of $\mathbb R^4$ such that $U ∩ W = \{{0}\}$ and $\dim(U) + \dim(W) = \dim(\mathbb{R^4})$. ...
4
votes
2answers
63 views

On chain complex morphisms

The following seems quite obvious to me. Nevertheless I would like to have another opinion. Suppose $(A_\bullet,d_A)$ and $(B_\bullet,d_B)$ are chain cmplexes, such that $d_A$ is the trivial ...
1
vote
0answers
77 views

Problem generating random vectors with a randomized linear programming with equality constraints (weird clustering)

Summary For simulation problems, I need to be able to generate large numbers of random lists of numbers, say $x_1, x_2, \dots, x_n$ (where $n \approx 1000$), subject constraints similar to what one ...
3
votes
1answer
67 views

$\|x -y\|+\|y-z\|=\|x-z\|$ implies $y= a x + b z$ where $a +b =1$

$\|x -y\|+\|y-z\|=\|x-z\|$ implies $y= a x + b z$ where $a +b =1$ Hint: Take $m=x-y$ and $n= y-z$. Does this follow from standard properties of inner product spaces (linearity, symmetry, and ...
0
votes
0answers
47 views

interpolation and Vandermonde

Looking at a problem of interpolation, I find a Vandermonde type matrix. To be precise I consider the following, let $$A(z)= \sum_{i=1}^p \sum_{j=2}^{n_i+1}\frac{a_{i}^j}{(z-z_i)^j}$$ where the $z_i$ ...
9
votes
2answers
232 views

Show $A$ and $B$ have a common eigenvector if $A^2=B^2=I$

Let $n$ be a positive odd integer and let $A,B\in M_n(R)$ such that $A^2=B^2=I$. Prove that $A$ and $B$ have a common eigenvector.
1
vote
0answers
231 views

Error on optimization problem, maximize log determinant on CVX

$A$ is an $N \times N$ complex matrix $W$ is an $N \times N$ complex matrix $C$ is an $N \times N$ complex diagonal matrix $u$ is a scalar $V$ is an $N \times N$ complex matrix, whose diagonal elects ...
0
votes
2answers
970 views

About the relation of rank(AB), rank(A), rank(B) and the zero matrix

Let $A$ be a $2 \times 4$ matrix and $B$ be a $4 \times 4$ matrix, prove that if $rank(A) = 2$ and $rank(B)=3$ then $AB \neq 0$. I got stuck at $rank(AB) \leq 2 $ How do I continue from here?
0
votes
2answers
250 views

triangular matrix and linearly independence

Let $T$ be a triangular matrix where $t_{ii}\ne 0$ for all $i$. Show that the rows and the columns of $T$ are linearly independent. I think it is obvious from the structure of $T$. But I do not know ...
2
votes
0answers
51 views

eigenvector and operator [duplicate]

Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, let $T:V\rightarrow V$ be a linear operator, and let $M$ be a nontrivial subspace of $V$ that is invariant under $T$. Prove that ...
0
votes
1answer
141 views

How to simulate IMU data using position and orientation?

I want to make a simulator to verify that my imu algorithm is working. I am given: $p_0$ - starting position $p_1$ - final position $q_0$ - starting orientation $q_1$ - final orientation I want ...
3
votes
2answers
176 views

How can a group of matrices form a manifold?

So for example, $GL(n,\mathbb{R})$ group. It is said that this group can be considered as manifold - but I do not get how this is possible. How does one then assign a neighborhood of a matrix, and ...
5
votes
7answers
935 views

Proving the relation $\det(I + xy^T ) = 1 + x^Ty$

Let $x$ and $y$ denote two $n$ - length column vectors. Prove that $$\det(I + xy^T ) = 1 + x^Ty$$ Is Sylvester's determinant theorem an extension of the problem? Is the approach same?
-1
votes
1answer
109 views

Prove that $A$ is Symmetric matrix or antisymmetric matrix

let matrix $ A\in R^{n\times n},\alpha\in R^{1\times n}$ if $\forall \alpha, \exists k\in Z $,then we have $A\alpha =kA^T\alpha ,$ prove that $A$ is Symmetric matrix or antisymmetric matrix
1
vote
1answer
95 views

Proving the existence of integer solutions to linear inequalities

Let $b_k\in\mathbb{Z}^n$ for $1\le k\le m$ for some $m,n$. I wish to prove the existence of two vectors $x,y\in\mathbb{Z}^n$ such that for all $k$, $b_k\cdot x\ne 0$ and $b_k\cdot y\ne 0$ with ...
8
votes
5answers
4k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
0
votes
2answers
103 views

Orthogonality of eigenvectors of laplacian

Let $x_i=(\sin i\pi/n,\cdots,\sin (n-1)i\pi/n)$ for $i=1,\cdots,n-1$. I want to show that $x_i \cdot x_j=\delta_{ij} n/2$. Why is it true? I tried $\sin a \sin b=-[\cos(a+b)-\cos(a-b)]/2$ but don't ...
3
votes
2answers
259 views

Jacobi Method and Frobenius Norm Question.

I have this linear algebra question concerning the jacobi method and the frobenius norm that I am having a lot of trouble on, I have an exam soon and I would appreciate any help. NOTE: I have read the ...
1
vote
1answer
153 views

$\operatorname{rank}AB\leq \operatorname{rank}A, \operatorname{rank}B$ [duplicate]

Prove that if $A,B$ are any such matrices such that $AB$ exists, then $\operatorname{rank}AB \leq \operatorname{rank}A,\operatorname{rank}B$. I came across this exercise while doing problems in ...
0
votes
1answer
74 views

Can we conclude that $\text{Im}\;E\subset\left(M+M^\perp\right)$?

In the above statement, $M=\ker E$. I asked one question (here) like this, but now I added one hypothesys. Could someone help me again? Notation: $V$ is a infinite-dimensional inner product space; ...
1
vote
5answers
353 views

Show that the set of polynomial functions is not finitely spanned

The set of the polynomial functions $P(\Bbb R)$ is a subset of functions in the form of $$f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n.$$ I wish to show that the set of polynomial functions is not finitely ...