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 ...

learn more… | top users | synonyms

0
votes
1answer
39 views

What is the best way to buy this food?? Please help, have the basics.

Prices:2016 |14 Meals| 28meals | 60 meals | 120 meals | 240 meals | $74 $148 $222 $296 $370 Throughout 2 years I ...
0
votes
3answers
201 views

Orthogonal Transformations

How can I show that given any two unit vectors in Euclidean space, there is an orthogonal transformation taking one to the other? I considered something like a reflection, but I don't know how to ...
0
votes
1answer
68 views

Subspace $\ell^2$ of square-summable sequences

How do I check that a vector $x=(1, 1/2, 1/4, 1/8,...)$ is in the subspace $\ell^2$ ? Thanks for any help!
0
votes
1answer
28 views

Vector Spaces V and W with linear transformation T: V --> W statements

I'm in a little slump with this question. I have a general idea, but I don't know exactly which theorem to pair them up with because I think that it may be too simple. Here is the question: For ...
3
votes
3answers
357 views

Determinant of matrix $A^3 + 2A^2 - A - 5I$ Given the eigenvalues of A

So A is a 3 by 3 matrix with eigenvalues -1, 1, 2. And I have to find the determinant of $$A^3 + 2A^2 - A - 5I$$ Let $u$ be the eigenvector for the eigenvalue -1. Let $S = A^3 + 2A^2 - A - 5I$ then ...
0
votes
1answer
69 views

Find the transformation that converts a square with diagonal vertices (0 , 3) and (-3 , 6) into a unit square at the origin.

Find the transformation that converts a square with diagonal vertices (0 , 3) and (-3 , 6) into a unit square at the origin.
3
votes
4answers
212 views

What's a good reference to study multilinear algebra?

This semester I'm taking a course in linear algebra and now at the end of the course we came to study the tensor product and multilinear algebra in general. I've already studied this theme in the past ...
1
vote
2answers
378 views

Proving that a set is an orthonormal basis

Any ideas on how to quickly show that $$ \left( \frac{1}{\sqrt{2\pi}}, \frac{\sin(x)}{\sqrt{\pi}}, \frac{\sin(2x)}{\sqrt{\pi}}, ..., \frac{\sin(nx)}{\sqrt{\pi}}, \frac{\cos(x)}{\sqrt{\pi}}, ...
1
vote
2answers
307 views

how covert joysitck (x, y) coordinates to robot motor speed?

I am trying to formulate an equation to calculate left and right motor speeds of a robot. The robot is four wheeled drive with independent motors connected to tank-like best on each side. see this ...
1
vote
0answers
99 views

Irreducible modules - semisimple algebras and endomorphism rings

Let $A$ be a finite dimensional, semi-simple $k$-algebra and $V$ and irreducible $A$-module. I am trying to prove the following claim: If $B = \text{End}_A(V^{\oplus r})$ then $$W = ...
0
votes
1answer
22 views

Inverse of $\{a_1 A_1,…,a_n A_n\}$

$a_1,...,a_n\in \mathbb{R}$ $A_1,...,A_n$ are the rows of the invertible matrix A I am trying to find a regular formula for this. Is it possible? Thanks for help!
0
votes
6answers
130 views

Prove or disprove: $ A^2 = I \Longrightarrow A=I \vee A=-I $

Linear Algebra/ Matrices A is in a $n\times n$ matrix. If $$ A^2 = I $$ does this imply: $A=I \vee A=-I $ Thanks!
0
votes
2answers
111 views

Calculating the matrix corresponding to linear map

How does one go about converting a linear map in functional form to a matrix; for instance: For a fixed unit vector $\hat{n} \in \mathbb{R}^{3}$, define the map $f:\mathbb{R}^{3}\to\mathbb{R}^{3}$ ...
1
vote
1answer
55 views

projection of inner products

Update of question Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define our inner product to be: $$ \langle p,q\rangle = ...
1
vote
0answers
144 views

eigenvalues with strictly negative real parts

$\textbf{Question: }$ If all the eigenvalues $\lambda_i$ of an $n\times n$ matrix $A$, have a strictly negative real part then prove that all the coefficients $a_j$ of the characteristic polynomial ...
1
vote
1answer
155 views

Historical reason to define a matrix vector product the way it is

What is the reason why we defined a matrix vector product (a transformation) this way: $$\begin{pmatrix} a_1 & a_2 \\ a_3 & a_4 \\ \end{pmatrix}\cdot ...
4
votes
1answer
97 views

Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$

I don't know how to prove the following: Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$, where $A \in GL_2(\mathbb{R})$, $b$ is a fixed vector in ...
0
votes
1answer
74 views

Matrix one-to-one and onto

I have the standard matrix \begin{array} {lcr} 0 & -2 & 0 \\ 1 & 0 & -1 \\ -1 & 2 & 3 \\ \end{array} I know that the reduced form is \begin{array} {lcr} 1 & 0 & 0 ...
1
vote
0answers
19 views

elements of oriented matroids belonging either to positive circuits or positive cocircuits

I need to prove the following, which seems trivial because it follows from the Farkas lemma (you may know this as the 3 or 4 painting lemma). Can someone show me how to prove this, please? I'm a bit ...
3
votes
1answer
42 views

Transformation of matrix - kernel

Let $T: \mathbb R^3 \to \mathbb R^2$ be given by $T(x,y,z) = (y,z)$. Show that $T$ is linear. Prove that the $\ker(T) = \mathrm{Span}(e1)$. Could you help me?
3
votes
1answer
89 views

Solution to linear system of equations

Notation. Let $y$, $a$, and $b$ be $n\times 1$, $p\times 1$, and $q\times1$ real vectors. Let also $X$ and $Z$ be $n\times p$ and $n \times q$ real matrices. Suppose that there is no solution, $a$, ...
1
vote
1answer
292 views

Finding a genterating set for the null space of T

I am not really sure what my book is talking about, they didn't talk about any of this until this practice problem. The standard matrix of T is \begin{array} {lcr} 1 & -1 & 2 \\ -1 & 1 ...
1
vote
1answer
55 views

Linear Space and Inner Products

Prove that the set $$ V=\{f\quad|\quad f:\mathbb{R} \to \mathbb{R}, f \quad\text{is absolutely integrable over} \quad \mathbb{R} \} $$ is a linear space over $\mathbb{R}$. Is it necessary to go over ...
0
votes
1answer
388 views

Proof of an elementary property of Projection Operators

I'm asked to show the following: Let $X$ be a linear space, and let $P : X \rightarrow X$ be a projection operator. Restricted to the linear space $range(P)$, the projection $P$ is the identity ...
5
votes
1answer
217 views

Is $(tr(A))^n\geq n^n \det(A)$ for a symmetric positive definite matrix $A\in M_{n\times n} (\mathbb{R})$

If $A\in M_{n\times n} (\mathbb{R})$ a positive definite symmetric matrix, Question is to check if : $$(tr(A))^n\geq n^n \det(A)$$ What i have tried is : As $A\in M_{n\times n} (\mathbb{R})$ a ...
0
votes
1answer
35 views

Avoid evaluation of a very large matrix in non-negative matrix factorization

This is somewhere in between a math and a programming question, so please send me back to SO if you think it's off-topic. I'm implementing non-negative sparse coding, a regularized variant of ...
1
vote
2answers
575 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
0
votes
2answers
31 views

Is $A$ s.t $A_{i, j} = x^T_i x_j$ semi-positive definite?

Let $x_1, x_2, \ldots, x_k \in \mathbb{R}^n$ and set define a $k$ by $k$ matrix $A$ by setting $A_{i, j} = x^T_i x_j$. Is $A$ semi-positive definite? If so, how can I show it?
1
vote
1answer
72 views

Finding a vector that is in two subspaces

The following question has been posed to me: You are given two subspaces $U$, $V$, both in $\mathbb{R}^n$, each with a basis of column vectors forming the columns of the respective matrices $U$, $V$. ...
2
votes
0answers
135 views

Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$ \Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}. $$ Applying the Woodbury matrix identity ...
0
votes
3answers
42 views

Non-Homogeneous System [Problem]

"Find a general solution of the system and use that solution to find a general solution of the associated homogeneous system and a particular solution of the given system." $\begin{bmatrix}3 & 4 ...
2
votes
1answer
148 views

How to find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ satisfying $f(XY)\le\min{(f(X),f(Y))}$

Let $M_n(\Bbb R)$ be the set of all real $n\times n$ matrices. Find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ such that $$f(XY)\le\min{(f(X),f(Y))}$$ for all $X,Y\in M_n(\Bbb R)$. ...
6
votes
3answers
248 views

Integer Programming problem

I have an integer programming problem with $L$ variables $x_1, x_2, x_{L}$ which all assume integer values and the following constraints must stand: $x_i \geq 0$ $x_1 = 10$ $x_2 + x_3 + ... + x_{L} ...
0
votes
1answer
72 views

Finding the “middle 2” of four lines

This may seem like an overly abstract problem, but it's the best generalization I could make of a specific problem I'm trying to tackle. This problem works in 2-dimensional Euclidean space. A ...
0
votes
2answers
127 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
2
votes
2answers
64 views

Is there an axiomatic definition of the concept “field equipped with a conjugation operator”?

In some sense, $\mathbb{C}$ is more than just a field, since aside from the usual field operations, it is also equipped with a conjugation operator $\mathbb{C} \rightarrow \mathbb{C}$. This means a ...
2
votes
1answer
79 views

Is there a non trivial ideal for the set of upper triangular matrices?

Is there a non trivial ideal for the set of upper triangular matrices? the zero matrix is a trivial ideal. Also, the set of upper triangular matrices is an ideal. Are there any other ideals?
0
votes
3answers
62 views

Inner Product of Real Polynomials

Updated improved question: Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define $$ \langle p,q\rangle = ...
0
votes
1answer
30 views

Uniqueness of Function (inner product?)

Consider a bilinear function $f : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is symmetric, i.e. $$ f(A_1, A_2) = f(A_2, A_1) \quad\forall A_1, A_2 \in \mathbb{R}^n $$ and satisfies $$ ...
3
votes
3answers
592 views

Notation: subscript vs. superscript for coordinate vector fields

Some books write the coordinate vector fields with a subscript as $$\frac{\partial}{\partial x_i}$$ while some write it with a superscript as $$\frac{\partial}{\partial x^i}.$$ Is there a conceptual ...
2
votes
1answer
152 views

expectation of norm of orthogonal projector

The question has to do with calculating the expected squared norm of a random projection. We have a 2D subspace $T := span\{U1, U2\}$ where $U1$ is a random vector uniformly distributed over unit ...
1
vote
1answer
54 views

real polynomials

Improved part to this question Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define $$ \langle p,q\rangle = ...
0
votes
2answers
140 views

angles of polynomials

Here is an improved question that was asked before. Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Let our inner product be defined by: $$ \langle ...
1
vote
1answer
350 views

Determine homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $y = 2x – 6$

Determine the homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $ y = 2x – 6$. I do $mx - y +b =0$: $\text{slope} = m$, $\tan(O)= m$ ...
0
votes
1answer
70 views

Linear density variation on a 2D plane

Here's my problem: on a 2-dimensionnal plane, I know the $x$, $y$ coordinates of 3 points $A, B, C$ each point comes with an associated density $T$, that can vary from infinite minus to positive ...
1
vote
2answers
98 views

Prove that $(V \oplus W) / (X \oplus Y)$ is isomorphic to $(V / X) \oplus (W / Y)$

Let V and W be two vector spaces over the field F. Let X ⊆ V and Y ⊆ W be subspaces. I don't understand how isomorphism figures here. Any help would be appreciated.
2
votes
2answers
95 views

Equation of a line through a point on a plane

Find the line that passes through the point $(2, 5, 3)$ and is perpendicular to the plane $2x - 3y + 4z + 7 = 0$ My only real problem with this is how to shift the line My first step is to find the ...
2
votes
2answers
34 views

Homogeneous Equations and Such

"Consider the linear system $\begin{bmatrix} 1 & -2 & 3\\2 & 1 & 4\\1 & -7 & 5\end{bmatrix} * \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ ...
0
votes
3answers
167 views

A finite dimensional vector space that is not naturally isomorphic to its dual.

I need an example of a finite dimensional vector space $V$ that is not naturally isomorphic to $V^\ast$. I know that, at least in finite dimensional case, there is a one-to-one correspondence between ...
1
vote
2answers
68 views

Calculating a determinant

How do I calculate the determinant of the following matrix: $$\begin{matrix} -1 & 1 & 1 &\cdots & 1 \\ 1 & -1 &1 &\cdots &1 \\ 1 & 1 & -1 &\cdots &1\\ ...