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 ...

learn more… | top users | synonyms

0
votes
2answers
9 views

Inspecting vector linear dependence, one line in matrix all zeros

Suppose we have vectors $v1, v2,v3$ and we want to inspect their linear dependence. They are linearly dependent when the only solution for the equation $\alpha * v1 + \beta * v2 + \gamma * v3 = 0$ is ...
1
vote
1answer
12 views

Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$

Let $V$ & $W$ be two finite dimensional vector spaces over $R$ and let $T_{1}:V\to V$ & $T_{2}:W\to W$ be two linear transformations whose minimal polynomials are given by ...
0
votes
1answer
10 views

Diagonalizable and Invertible Functions

In each of the following parts, either give an example of a linear function T: C$^2$ -> C$^2$ with the specified properties (and show that your example has the desired properties), or prove that no ...
0
votes
0answers
5 views

Convert global 2D coordinates to local

I have some 2D coordinates [x,y] and an object whose position and rotation is represented by a 3x3 matrix, with the form: [1 0 x] [0 1 y] [0 0 1] I need to get ...
7
votes
1answer
59 views

Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; $\forall n\in \mathbb{N}$

$a;b;c;x;y;z \in \mathbb{R}$ such that : \begin{matrix} a+b+c=x+y+z & \\ a^2+b^2+c^2=x^2+y^2+z^2 & \\ a^3+b^3+c^3=x^3+y^3+z^3 & \end{matrix} Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; ...
0
votes
0answers
11 views

Find a basis for a subspace of an inner product

Consider the vector space P$_2$(C), with inner product defined by $\langle{p(x)}$,$q(x)\rangle$ = $\int_0^1{p(x)\overline{q(x)}}dx$ Let W = {p(x) $\in$ P$_2$(C) : p'(0) = 0}. You may assume, without ...
4
votes
1answer
17 views

Determine whether the set of vectors is linearly dependent or not

Suppose I have the vectors $\underline{a}_1, \underline{a}_2,\ldots,\underline{a}_k$ and $\underline{b} \neq 0$ in $\mathbf{R}^n$. Also, $\underline{a}_1 \neq \underline{a}_2 \neq \ldots\neq ...
0
votes
0answers
15 views

Why is $P_{E^\bot}(x)=0$, if $x\in E$?

Let $H$ be a Hilbert space and let $x\in H$. Let $E$ be a non-empty closed subspace of $H$. Let $P_E(x)$ be the projection of $x$ unto $E$. I've seen several proofs that use the following: ...
0
votes
1answer
12 views

Intersection of kernel of commuting nilpotent matrices

Suppose $N$ and $Q$ are two nilpotent matrices which commute. Is it true that $\ker N \cap \ker Q \ne \{ 0\}$?
0
votes
0answers
8 views

Prove the following equation is solvable in $M_m(\mathbb C)$.

Prove the following equation is solvable in $M_m(\mathbb C)$ for any $n, l, m \in \mathbb N^*$: $${X^n}+{X^l}-{I_m}=\left( \begin{matrix} 1 & 0 & 0 & \cdots & 0 \\ 2 & 1 ...
3
votes
2answers
33 views

Proving that the sum of elements of two bases is a basis

I am given a (not necessarily orthonormal) basis of a certain finite vector space $\{e_i\}_{1\leq i\leq n}$. Now, after the usual Gram-Schmidt orthonormalization procedure, I end up with an ...
0
votes
1answer
15 views

Finding the conditions of a system of equations for a type of solution

Consider the system of equations $x$,$y$, and $z$, $$2x+3y-z=p$$ $$x-2z=-5$$ $$qx+9y+5z=8$$ where $p$ and $q$ are real. Find the values of $p$ and $q$ for which this system has: ...
0
votes
0answers
16 views

Search Patterns. The Circling of The Circle and The Squaring of The Square.

Consider these two search patterns. {1} A square moving in straight lines in what you might call a "square-spiral" pattern, covering an infinitely large square. {2} A circle spiraling out covering ...
1
vote
1answer
30 views

Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?

It is known that any infinite cyclic group , in particular $(\mathbb Z, +)$ , can never be a vector space . So we may ask , Can we define an operation $*$ on $\mathbb Z$ such that $(\mathbb Z , *)$ ...
1
vote
1answer
25 views

show that $A(t)\exp(\int_{t_0}^t A(s)\,ds )=\left(\exp(\int_{t_0}^t A(s)\,ds )\right)A(t)$, when $A(t)$ is symetric.

$A(t)$ is a symetric matrix for $t\in [t_0,a]$. show that $$A(t)\cdot \exp\left(\int_{t_0}^t A(s)ds \right)=\exp\left(\int_{t_0}^t A(s)ds \right)\cdot A(t)$$ it is easy but exhausting to show for ...
1
vote
1answer
24 views

Prove that a matrix is invertible?

Let $A_{20 \times 20}$ be a real matrix such that: $\ \ \ \bullet$ $a_{ii}=0$ for $1 \le i \le 20$ $\ \ \ \bullet$ $a_{ij} \in \{-1;1\}$ for $1 \le i,j \le 20$ and $ i \neq j$ Prove that $A$ is ...
0
votes
1answer
24 views

Linear Algebra, Spans and subspaces

Let $V= \mathbb{R^3}$ and consider the following elements of $V$: $\mathbf{u}_1 =(1,2,0)$, $\mathbf{u}_2=(3,1,0)$, $\mathbf{u}_3=(1,-1,1)$. Let $U= \langle\mathbf{u}_1,\mathbf{u}_2\rangle$ and ...
2
votes
3answers
77 views

The number of the solutions of $‎ x^{10}=‎ ‎ ‎\begin{bmatrix}1&0\\‎ ‎0&1‎ ‎\end{bmatrix}‎$

How many solutions does the following equation have in $ M_{2}(\mathbb R)$ and why? $$‎ x^{10}=‎ ‎\begin{bmatrix}1&0\\‎ ‎0&1‎ ‎\end{bmatrix}‎$$ Every hint is appreciated.
4
votes
0answers
21 views

Characterizing $\text{PGL}_2(\mathbb F_p)$

Where can I find a description and proof of the basic structure of $\operatorname{PGL}_2(\mathbb{F}_p)$ (Number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
0
votes
0answers
8 views

Singular Jacobian in Newton's method

How can we prove that Newton's method for a non-linear system converges linearly (as opposed to quadratically) if the Jacobian is singular at the root? Is this related to being multiple roots at that ...
3
votes
2answers
83 views

Can the zero vector be an eigenvector for a matrix?

I was checking over my work on WolfRamAlpha, and it says one of my eigenvalues (this one with multiplicity 2), has an eigenvector of (0,0,0). How can the zero vector be an eigenvector?
0
votes
2answers
38 views

Can a $3\times3$ matrix have more than $3$ linear independent eigenvectors?

I understand you can do multiples of eigenvectors, but suppose they are a linear independent. Can there be more than $n$ for a $n\times n$ matrix?
0
votes
2answers
19 views

Parametric / vector question.

Question 10 [10 points] Let L be the line with parametric equations $$ x = −6−3t $$ $$ y = 6+3t $$ $$ z = −8+2t $$ Find the vector equation for a line that passes through the point P=(−1, 2, 3) and ...
0
votes
0answers
39 views

A basic doubt to compute exponential of a matrix

Given a matrix I want to evaluate $e^{A}$. The method suggested uses the taylor expansion. But, it is also written that the method works well if the largest and smallest eigen values are not well ...
1
vote
1answer
15 views

How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...
0
votes
0answers
23 views

Does this guassian elimination have a solution?

I was asked to find the following solutions using guassian elimination, but I was unsure of my answers since it became quite messy but the variables still somehow fit: $$\left[\begin{array}{ccc|c} ...
0
votes
0answers
21 views

The expected value of a random vector when the X_is are independent

$ \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cov}{cov} $ The components of a random vector $\mathbf{X} = [X_1, X_2, \ldots, X_N]^{\intercal}$ all have the same mean $E_X[X]$ and the same ...
-1
votes
0answers
22 views
0
votes
1answer
25 views

Prove True or false : If A and B are nxn invertible matrices and (AB)^2=A^2B^2, then AB=BA

This looks like it is false but the thing is I can't find a counter example for it.
1
vote
1answer
26 views

Fixed field of two subgroups of $\operatorname{Aut}_{K}{K(x)}$

This link explains $\operatorname{Aut}_{K}{K(x)}$. And I want to know how to solve two problems below in the Hungerford's Algebra, p.256. $7.$ Let $G$ be the subset of $\operatorname{Aut}_{K}{K(x)}$ ...
0
votes
2answers
7 views

Matrix Norm Division

Suppose $A=uv^*$ where $u$ is an $m$-vector and $v$ is an $n$-vector. For any $n$- vector $x$, we can bound $||Ax||_2$ as follows: $||Ax||_2 = ||uv^*x||_2=||u||_2|v^*x|\leq||u_2||||v||_2||x||_2$. ...
1
vote
1answer
18 views

Number of positive, negative eigenvalues and the number of sign changes in the determinants of the upper left submatrices of a symmetric matrix.

How do we prove that the number of sign changes in the sequence of the determinants of the upper-left matrices of a symmetric matrix $A$ corresponds to the number of positive and negative eigenvalues ...
0
votes
0answers
20 views

linear algebra find a line that intersects another line

question: Let L be the line with parametric equations x = 3+2t y = −5 z = −6−3t Find the vector equation for a line that passes through the point P=(−5, 5, −6) and intersects L at a point that is ...
1
vote
1answer
25 views

Sets of binary sequences

In my course on linear algebra we have recently introduced linear independent subsets of vector spaces. As an exercise I have been thinking about examples of infinite linearly independent sets and ...
0
votes
2answers
32 views

Does basis of eigenspace mean the same as eigenvectors?

If you have a 3x3 matrix, 2 eigenvalues (one with multiplicity 2) and now 2 eigenvectors, how do you find the basis for each eigenspace?
1
vote
2answers
18 views

Orthogonal Matrices and Similarity Transforms

Sorry I can't be more specific with the title. I really don't know what to call this and about 2 hours of Googling has yielded no results. All we are given: $U$ is $n\times n$ and orthogonal $Ax = ...
1
vote
0answers
23 views

Can the number of sign changes in a sequence of determinants tell us how many negative eigenvalues a symmetric matrix has?

From notes, I've gathered that given a symmetric matrix, the number of sign changes in its characteristic polynomial is equal to the number of positive eigenvalues of $A$. Proof: Let $p(x)$ be a ...
0
votes
1answer
17 views

Uniqueness of Thin QR Factorization.

Let $A \in \mathbb C^{m x n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m x n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal elements, ...
0
votes
1answer
33 views

Simultaneous function with three variables using subsititution method

Use any substitution method and solve the following equations: $$2x+5y+7z=86$$ $$3x+y+5z=60$$ $$x+4y+3z=54 $$ I used $x+4y+3z=54$ to make $x$ the subject $x=54-4y-3z$.
0
votes
1answer
12 views

Covariance matrix of Y when we have the covariance matrix of X

If the random vector $\mathbf{X}$ is transformed according to \begin{align*} Y_1 &= X_1\\ Y_2 &= X_1 + X_2 \end{align*} and has a covariance matrix $$ \mathbf{C}_X = ...
1
vote
2answers
33 views

Does the line $(2,1,1)+t(-3,1,5)$ live within the plane $31x+3y+18z=62$?

I have a doubt with this exercise: Have the plane $$31x+3y+18z=62$$ What is the distance between this plane and some line $(x,y,z) = (2,1,1) + t(-3,1,5)$ for some $t\in\mathbb{R}$? The ...
1
vote
0answers
38 views

Differences of grade between this three books

I have a simple question, I noticed these three books for my study, but I didn't understand the grade of these books because the names of the paragraphs are similar . 1) ...
3
votes
1answer
25 views

For what kind of matrix does it hold $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
3
votes
1answer
19 views

Rank of a special matrix

Say a $5\times 5$ matrix $$A = \left[ \begin{array}{ccc} 1&2&3&4&5\\ 6&7&8&9&10\\ 11&12&13&14&15\\ 16&17&18&19&20\\ ...
1
vote
2answers
25 views

How to show that $\| QA\|_2=\| A \|_2$ where $Q$ is unitary (for a matrix A)

I want to show that for a unitary matrix $Q$ and a matrix $A$ that $$ \|QA\|_2=\|A\|_2$$ I start with the definition of matrix induced norms: $$\| QA \|_2 = \sup_{x \neq ...
0
votes
0answers
24 views

How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?

How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?
2
votes
2answers
27 views

Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
0
votes
2answers
22 views

Number of Jordan canonical form of a matrix

Let, $A\in M(3,C)$. Assume that the characteristic & minimal polynomial of $A$ are known. Then what is the number of possible Jordan form of $A$ and how? What changes if we replace $C$ by $R$ or ...
0
votes
0answers
12 views

Eigensystem of a real symmetric Toeplitz matrix of large order

My question is related to this one. I am looking for the eigenvalues and eigenvectors of a square, symmetric, real Toeplitz matrix of order N where N is large. There are some references in the above ...
0
votes
1answer
30 views

Are the following quotient spaces finite dimensional?

If we take $\mathbb{F}[x]$ to be the set of all polynomials over the field $\mathbb{F}$, $E$ to be the subset of all such even polynomials, $N$ to be the set of these polynomials that have degree less ...