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Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

2
votes
1answer
15 views

Does a linear change of coordinate change the reducibility of a polynomial?

Let $R$ be an integral domain, $f$ $\in R[x]$, is it true that $f(rx+b)$ being irreducible implies that $f$ irreducible for $r,b \in R$? I know that this is true for $R= \mathbb R$ or $\mathbb C$. ...
0
votes
0answers
8 views

Write $\frac{(X^{2} + Y^{2})}{ XY}$ in Quadratic Form

Is there a way to write $$ \frac{X^{2} + Y^{2}}{XY} $$ in quadratic form. I am struggling to set up the proper vector and find a square matrix.
-1
votes
1answer
8 views

Algebraic representation of Matrix product

Suppose $M = X^T \Delta X$, where $X$ and $\Delta$ are $P \times P$ matrices and $\Delta$ is symmetric and positive-definite. Can anyone give a simple algebraic representation of the matrix $M$?
2
votes
1answer
27 views

How is it true? if a polynomial has no solutions then it is irreducable.

So I found this statement in my friend's notes and I think it's wrong for example if $x^2 + 1 = 0$ has no solution and is not reducable but if I square the whole thing then it is reducable but still ...
0
votes
1answer
8 views

When product of matrix and vector equals 0, why does product of matrix, it's transpose and vector also equals 0?

I've got a matrix A and a vector s When Av = 0, why does A*As = 0 and also s*A*As = 0? A* is transposed matrix A.
0
votes
1answer
8 views

Eigenvalues of a block off-diagonal matrix

Let $A_1,A_2 \in \mathbb{R}^{n \times n}$. Construct the block matrix $A$ as follows: $$A: = \left[ {\begin{array}{*{20}{c}} 0&{{A_1}}\\ {{A_2}}&0 \end{array}} \right]$$ My observation is that ...
1
vote
0answers
24 views

Showing 2 matrices are similar

I gotta show if or if not those 2 matrices are similar: $$ \left(\begin{matrix} 3 & 2 & -2 \\ 1 & 4 & 0 \\ -2 & 1 & -1 \\ \end{matrix}\right) $$ $$ \...
3
votes
1answer
16 views

Find rank of AB, given that A has linearly independent columns and B has rank 2

I'm trying to prove to myself that given... Matrix A, which has linearly independent columns, and at least 2 columns... Matrix B, which has rank of 2 Their product, AB, will have rank of 2. I ...
1
vote
0answers
13 views

If union of n subspaces of V is a subspace of V, then one of the n subspaces must contain the other n-1 subspaces

Help prove this? I can prove for n=2, but I'm stuck on proving it for general n. Thanks! My proof for n=2 Forward direction: Consider A and B and A $\cup$ B is a subspace of V. We prove by ...
0
votes
2answers
22 views

transition from one base to another

Given the polynomials $p_1,p_2,p_3,v_1,v_2,v_3 \in P_2$: \begin{gather*} p_1(t) := t^2 − 2t + 5, \qquad p_2(t) := 2t^2 − 3t, \qquad p_3(t) := t + 1, \\ v_1(t) := t^2 + 4t − 3, \qquad v_2(t) := t − 1,...
1
vote
0answers
7 views

Finding ch. polynomial, min. polynomial and the Jordan n. form of $f$ knowing $f^3=0$ and $f(v_1)=f(v_2)=v_3, f(v_3)=kv_4, f(v_4)\in<v_1,v_4>$

Given a vector space $V$ of dimension $4$ and a base $\{v_1,v_2,v_3,v_4\}$, let $f$ be an endomorphism of $V$ such that $f^3=0$ and $f(v_1)=f(v_2)=v_3, f(v_3)=kv_4, f(v_4)\in<v_1,v_4>$, where $k$...
0
votes
2answers
9 views

Procedure to find a pair of matrices whose product has rank less than each matrix in the pair

I am trying to figure out how to find two rank-deficient matrices (not necessarily square) which when multiplied will have rank less than either of the original matrices. In other words I am looking ...
0
votes
2answers
23 views

Linear operator statement

I'm stuck with a practice problem question: '' Is it true that a linear operator, T : V → V, on one-dimensional vector space V over field F has the form T(v) = av, for all v ∈ V and a a scalar from ...
5
votes
1answer
37 views

Equality concerning the norm of rows of a resolvent matrix.

This problem showed up on UCLA's basic exam for Fall 2018: Let $X$ be an $n \times n$ symmetric (real) matrix and $z \in \mathbb{C}$ with $\text{Im } z > 0$. Define $G = (X - zI)^{-1}.$ Show ...
-1
votes
0answers
15 views

Parameterisations intuition

If I paramaterise a function like x= f(t) and y=g(t) like x=t and y=t^2 why is it that the intersection in the x,y,t plane, I only get a point? not y=$x^2$? I understand the method of eliminating the ...
-4
votes
2answers
38 views

$\text{Show that }F^{I}=F^{I}_{\text{fin}} \iff I \text{ finite, where }I \text{ is a set and } F\text{ is a field}$

$F^{I} \text{ is the vectorspace of functions which map from }I \text{ to } F \text{ and }F^{I}_{\text{fin}}:= \{ f:I\rightarrow F |\text{ } \overset{\exists}{n\in\mathbb{N}_0,}\overset{\exists}{I_f\...
1
vote
0answers
11 views

How to express a quadric equation from canonical form to a different basis.

I have the quadric $3X^2-Y^2-Z^2=0$ expressed in the canonical form, and the matrix of change of basis from a basis B to the canonical form is $$P=\begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\...
0
votes
1answer
825 views

When do the solutions to the linear system $Ax=b$ form a vector subspace?

When do the solutions to the linear system $Ax=b$ form a vector subspace? A) If and only if $A$ is invertible; B) if and only if $b=0$; C) if and only if $A$ is not invertible; D) if and ...
-1
votes
1answer
24 views
-5
votes
1answer
25 views

LINEAR ALGEBRA (LINEAR INDEPENDENT OR DEPENDENT) [on hold]

how to check vector: $(1,i,0), (2i,1,1),(0,1+i,1-i)$ are linearly inpendent or dependent.
3
votes
2answers
125 views

Count the number of bases of the vector space $\mathbb{C}^3$.

Problem: Consider the set of all those vectors in $\mathbb{C}^3$ each of whose coordinates is either $0$ or $1$; how many different bases does this set contain? In general, if $B$ is the set of all ...
1
vote
1answer
31 views

A matrix equation equivalence

Let $\Omega_{\, m\times m}$ be a real square positive definite symmetric matrix, $u_{m\times 1}$ is a vector, $I_{m\times m}$ is the identity matrix. Let $x$ be a solution of a matrix equation $$ u^...
1
vote
1answer
15 views

Locally trivial line bundle

I am reading the book "An Introduction to Contact Topology" by Geiges. In the proof of Lemma 1.1 where he proves that any co-dimension 1 hyperplane field distribution is locally kernel of a 1-form, ...
0
votes
1answer
14 views

If $B$ is a basis of $V$ and $U\subseteq V$ is linearly independent then there exists a $C\subseteq B$ such that$ \tilde{B}:= U\dot\cup C$ is a basis

It is an application of Zorn's Lemma, It would help me a lot if somebody could explain one part of the proof that I did not understand Why can we describe every element of $B$ as a linear ...
-2
votes
4answers
45 views

Distance between the origin (0,0) and a line y = ax +b [on hold]

Derive a formula for the Euclidean distance between the origin $(0,0)$ and a line $y = ax + b$, where $a$ and $b$ are arbitrary constants.
0
votes
0answers
30 views

How to prove the limit $\lim_{k\rightarrow\infty}\lambda^{k-j}\binom{k}{k-j}=0,\; k>j,$ and $k\gt j$

I find difficulty proving the following limit: Suppose $|\lambda|<1$,then $$ \lim_{k\rightarrow\infty}\lambda^{k-j}\binom{k}{k-j}=0,\quad k>j. $$ When I try to prove that the $k$th Jordan ...
0
votes
0answers
11 views

Is working out the smith normal form trivial?

What's the point of using elementary column operations to work out the smith normal form of a matrix of reals when it is easy to work out exactly how many redundant equations there are from the row ...
2
votes
2answers
68 views

Subspace of a vector space problem

So the task says: for $\alpha_1\ldots,\alpha_n, \beta\in \mathbb R$, we define $U = \{(x_1\ldots,x_n) ∈ \mathbb{R}^n \,|\, \sum_{i=1}^n \alpha_ix_i = \beta\}$. When is the $U$ subspace? I know that ...
0
votes
0answers
27 views

How to choose $B$ to have $ \operatorname{trace}(H) \ge \operatorname{trace}(B^{\top}HB)$?

Considering $H \in \mathbb{R}^{N \times N}$ a real symmetric matrix which has both positive and negative eigenvalues, and $B \in \mathbb{R}^{N \times M}$ a real matrix with positive entries. Can I ...
0
votes
0answers
8 views

Eigenvalues of normalized vs unnormalized Laplacian of weighted digraph

Let $G$ be a weighted digraph. What is the connection between the eigenvalues of the normalized and unnormalized Laplacians of $G$. I think there is no explicit connection. We can at most find some ...
1
vote
1answer
20 views

Subspace of $ P_5$?

Problem: Let $U$:= {$p$ ∈ $P_5$$(\mathbb{R})$ : $p(−1)$ = $p(1)$ = $0$}.Show that $U$ is a subspace of $P_5$$(\mathbb{R})$ . Find a basis and determine the dimension of $U$. Solution: ...
1
vote
1answer
18 views

Equality of Frobenius Inequality

Consider the case when $$\DeclareMathOperator{\Rank}{Rank}\Rank(AB) + \Rank(BC) = \Rank(B) + \Rank(ABC)$$ where $A \in M_{m,k}(F)$ , $B \in M_{k,p}(F)$ and $C \in M_{p,n}(F)$ for any field $F$. I ...
1
vote
1answer
18 views

Quaternion product of three vectors: meaning of vector part?

$\newcommand{\i}{\mathbf{i}} \newcommand{\j}{\mathbf{j}} \newcommand{\k}{\mathbf{k}} \newcommand{\a}{\mathbf{a}} \newcommand{\b}{\mathbf{b}} \newcommand{\c}{\mathbf{c}} \newcommand{\R}{\mathbb{R}}$If ...
0
votes
1answer
18 views

On what condition $trace(A) \ge trace(AB)$?

Considering $A$ and $B$ are positive semidefinite real symmetric matrices, on what conditions we can have $trace(A) \ge trace(AB)$?
1
vote
0answers
31 views

Finite-dimensional representations of the integers (2)

I was reading the answer of the question from this link: Finite-dimensional representations of the integers But I have 2 things that I do not understand in the solution and the comments of the ...
1
vote
1answer
870 views

Linear algebra: Finding the number of parameters of a system given dimensions and rank

Suppose the system Ax = b is consistent and A is a 6 x 7 matrix and rank(A) = 2. How many parameters does the system have?
0
votes
1answer
23 views

A matrix equation involving eigendecomposition

Let $p<n$ and -$H\in\mathbb{R}^{n\times n}$ be symmetric with eigendecompoistion being $H=U\Lambda U^{\text{T}}$, -$A\in\mathbb{R}^{n\times p}$, -$D\in\mathbb{R}^{p\times p}$ be a diagonal ...
0
votes
1answer
15 views

Eigenvalue of an operator implies eigenvalue of the dual?

I helped some students today with linear algebra, which I took last year. They asked me a question from their homework to which I couldn't find an answer: Let $V$ be a finitely generated vector space ...
1
vote
1answer
73 views

Let $A$ be complex matrices and $x$ be eigenvalue of $A\overline{A}$.

Let $A$ be non-singular $n\times n$ complex matrices and $x$ be negative eigenvalue of $A\overline{A}$. Show that algebraic multiplicity of $x$ is even number. I show that the $\det(A\overline{A})>...
2
votes
1answer
25 views

Two matrices have the same eigenvalue …

This is an testexam question and that requires true or false and some explanataion. 2 (n x n) matrices have the same eigenvalue Pi, does A - B have eigenvalue 0?
0
votes
2answers
33 views

Notation and major confusion with the definition of the probability simplex

So it starts like this: Given a discrete set $N$, the probability simplex over $N$, denoted $∆(N)$ is defined to be: $$Δ(N) = \left\{ x \in \mathbb{R}^{|N|} \;:\; x_{i} \geq 0 \text{ for all } i, \...
1
vote
2answers
23 views

What is the significance of the fact that a set in a vector space is linearly dependent if it contains finitely many linearly dependent vectors?

A set $S$ in a vector space $V$ is linearly dependent if it contains finitely many linearly dependent vectors. What is the significance of this definition? Obviously for finite sets, this is no ...
1
vote
1answer
11 views

Visualization of 2-dimensional projective transformation

In analytic photogrammetry, there is a transformation called 2-dimensional projective transformation which makes a link between the map plane and the picture plane. If $(X,Y)$ is a point in the map ...
0
votes
0answers
26 views

Determining the rank, nullspace and image of a matrix with one unknown

Consider the matrix: $$A=\begin{pmatrix}x &0&2&2 \\-2&2&-2&2\\2&2&-2&2\\2&0&2&-2 \end{pmatrix}$$ Determine $\text{rank(A)}$ as well as a basis for the ...
1
vote
1answer
18 views

Question about inner product of $C_{2}[a,b]$.

My question, I think, is quite simple. I have a space $C_{2}[a,b]$ of complex-valued continuous functions. I checked that functional $$(f,g) = \int_{a}^{b} f(t)\overline{g}(t)dt$$ is a inner product ...
30
votes
2answers
4k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
6
votes
1answer
61 views

Prove all roots of $p_n(x)-x$ are real and distinct

Given a polynomial series $\{p_n(x)\}_{n=1}^{\infty}$ in $\mathbb{R}[X]$ with initial value $p_1(x)=x^2-2$. And $p_k(x)=p_1(p_{k-1}(x))=p_{k-1}(x)^2-2,\;k=2,3,\cdots$. Prove that for each integer ...
0
votes
0answers
22 views

(Solved) Question About Schur's Theorem

Schur's theorem is about Upper triangular representation of an linear operator. This is from Linear Algebra Done Wrong by Sergei Treil. (1) Schur's Theorem. Let $A: V \rightarrow V$ be an operator ...
0
votes
3answers
40 views

How can I solve the linear recurrence problem $f(n)=f(n-1)+3 \cdot f(n-3)+2n$ using matrix exponentiation when $ f(1)$ , $f(2)$ and $f(3)$ are given.

The porblem is $f(n)=f(n-1)+3f(n-3)+2n$. I solved $f(n)=f(n-1)+3f(n-3)$ and adding summation of $2n$ upto $n$. But this is wrong. It requires too much of pre calculation. I tried this problem based on ...
2
votes
0answers
89 views

Eigenvalues keep giving trivial solutions everytime.

I am trying to find the eigenvalues of this Eigen BVP. $\mu$ is the eigenvalue parameter $$ \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \...