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 ...

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-2
votes
0answers
17 views

System of Equations for 3-digit number [closed]

This is a rare word problem where I've had trouble: Find system of equations and use elimination. The sum of the digits of a three-digit number is 9, and the tens digit of the number is twice the ...
4
votes
2answers
56 views

Which $n$-forms are pullbacks of top forms on $\Bbb R^n$

Let $V$ be a finite-dimensional vector space. I write $F_n(V)$ for the $n$th exterior power of the dual vector space. Which elements of $F_n(V)$ can be pulled back from a top form along a linear ...
2
votes
1answer
25 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
-3
votes
1answer
54 views

Verify the following assertion: [closed]

Suppose that $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$.Then $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}.$$ Not sure how to add these subsets. Please provide explanation.
2
votes
1answer
68 views

How can I find $\det(A)/\det(B)$, when individual determinants blow up

I am interested in the quantity: $\frac{\det(A)}{\det(B)}$ of positive definite matrices $A$ and $B$. The problem I am running into now is that for large $A$,$B$, (around $200 \times 200$), the ...
-3
votes
0answers
20 views

How can I tell if two lines will cross using vectors [closed]

things I need 1. a visual recpinatation 2. A explanation on how to solve it and 3. a problem to do
1
vote
2answers
48 views

Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta ...
0
votes
1answer
87 views
+50

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
0
votes
0answers
25 views

Can I do Gaussian Elimination on this? (mod 2)

I have this matrix in GF(2): [0, 0, 1, 0] [1, 1, 0, 0] [0, 0, 0, 1] It's not a square matrix but I tried to do Gaussian elimination on it anyway after adding a ...
-1
votes
0answers
30 views

Don't understand about how to solve the first equation with Gauss Elimination

Please take a look at the picture. My quesion is: How to solve the first equation with Gauss Elimination that is displayed in the picture. I don't understand HOW.
0
votes
0answers
27 views

Linear independent vectors

I am stuck in an argument. Is the following true: Let $\{v_1,\ldots,v_k\}$ and $\{u_1,\ldots,u_k\}$ be sets of linear independent vectors. Set $u:=\sum_{i=1}^k \alpha_i u_i$, for some $\alpha_i$. Now ...
2
votes
1answer
61 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
2
votes
1answer
17 views

Determining dimension of a sum of subspaces in terms of a parameter

Problem: Consider the linear subspaces \begin{align*} U = \text{span} \left\{ (1,0,1,0), (1,a,0,a)\right\} \quad \text{and} \quad W = \text{span}\left\{(-1, a, a^2, 0), (0,1,0,-1)\right\} \end{align*} ...
10
votes
9answers
208 views

Show that $B$ is invertible if $B=A^2-2A+2I$ and $A^3=2I$

If $A$ is $40\times 40$ matrix such that $A^3=2I$ show that $B$ is invertible where $B=A^2-2A+2I$. I tried to evaluate $B(A-I)$ , $B(A+I)$ , $B(A-2I)$ ... but I couldn't find anything.
0
votes
0answers
18 views

How to compute the Eigenvectors for a Markov matrix?

I have the following matrix for which I want to get the Eigenvectors. I know how to compute the Eigenvalues, but when I compute the vectors in the null space of the matrix, I get the wrong answer. ...
3
votes
1answer
28 views

A linear functional on the space of transformations is basis independent

I've been working on this problem for a bit and am not sure how to proceed: let $V$ be an $n$ dimensional $\mathbb{ R } $-vector space, and denote by $\mathcal{L}(V)$ the space of linear operators $V ...
3
votes
8answers
89 views

For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?

While watching a video about dot products (https://www.youtube.com/watch?v=WDdR5s0C4cY), the following formula is presented for finding the angle between two vectors: For vectors $a$, and $b$, ...
2
votes
0answers
29 views

Eigenvalues of the product of two “incidence” matrices

I am trying to solve the following problem. Let the following incidence matrix of an undirected graph with four nodes $$ B = \begin{bmatrix}1 & 0 & -1 & 0 & 0 \\ -1 & 1 & 0 ...
3
votes
4answers
63 views

Does an $n\times n$ matrix $A$ only have an inverse if $rank(A)=n$? If so, why?

I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example ...
1
vote
1answer
23 views

On the projection onto the image set of an $m\times n$ matrix

I came accross as statement that: "If $K$ is the image set of an $m\times n$ matrix $A$ with full column rank, then $$P_Kx=A(A^TA)^{-1}A^Tx."$$ How do I verify this? I know that the inequality ...
1
vote
2answers
31 views

Solving system of nonlinear equations

Say I have a system of 4 equations, 4 unknown (A,B,C,D), how would you solve it analytically, assuming a, b, C1, C2, C3, C4, C5, C6, F, G, H, I are just some constants? If using Gaussian Elimination, ...
6
votes
2answers
39 views

minimum eigenvalue for difference of two matrices

Let $A$ a symmetric positive definite matrix, and $B$ a matrix constructed from $A$ by setting all its off-diagonal elements to zero. Then is there a way to see for which values of positive scalars ...
0
votes
4answers
49 views

Points on 3d line

Say we have $2$ points on a 3d line, point $A(x,y,z)$ and point $B(x,y,z)$. If we want to get the coordinates of a third point, beyond point $B$ but a certain distance from point $A$, how would we do ...
0
votes
1answer
22 views

Iterative solutions of linear systems

I do not understand that why $M$ must be invertible for $x^{(k+1)}$ to be uniquely specified in equation below: $$ Mx^{(k+1)} = Nx^{(k)} + b \quad (k=0,1,\ldots).$$ Why $M$ must be invertible? And ...
1
vote
2answers
27 views

a symmetric bilinear form has a basis such that it's matrix with respect to it is diagonal

I'm reviewing a proof regarding $f$, a symmetric bilinear form having a basis such that it's matrix with respect to this basis is diagonal. Here's a summarization: For $n=1$ there's nothing to ...
1
vote
1answer
14 views

Columns Of The Diagonalization Matrix

After finding the eigenvectors, we can create a matrix $Q$ such that $Q^{-1}\cdot A \cdot Q=D$ when $A$ is a matrix and $D$ is a diagonal matrix with the eigenvalues on the diagonal. In which order ...
3
votes
1answer
35 views

Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
0
votes
0answers
22 views

Eigenvalues of integrals over similar matrices

Let $\rho = \rho(x)$ be a $2\times2$ matrix (don't know if it is necessary, but $\rho$ is a density operator) and $I$ be the (two-dimensional) identity matrix. I have two matrices $A$ and $B$, where ...
1
vote
1answer
30 views

Prove there exists a self-adjoint transformation $C$ s.t. $CA=B$ if $A$ and $AB$ are self adjoint

If $A$ and $B$ are linear transformations such that $A$ and $AB$ are self adjoint and such that $Ker(A)\subset Ker(B)$, then prove there exists a self-adjoint transformation $C$ s.t. $CA=B$
1
vote
1answer
32 views

does the following matrix update law, converge? if so, to what?

Assume $A$ and $B$ to be arbitrary matrices (but you can assume some conditions on their norm), We have $X_{i+1}=AX_{i}A^T+B$, We are looking for $X_{\infty}$ (if it exists). Does it converge, if ...
0
votes
1answer
18 views

Characterising Adjugate(adjoint) of a matrix

If $A$ is an $n\times n$ matrix over a field, then adj$(A)$ is an $n\times n$ matrix (obtained from $A$) such that $$\mathrm{adj}(A)\,A=A\,\mathrm{adj}(A)=\mathrm{det}(A)I_n.$$ Question: If $B$ is ...
0
votes
1answer
51 views

Linear Algebra: Question about determinants

The following matrices are $4 \times 4$ matrices. $$A=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\\ 1 & 1& 1 &0\\ 1 &1 &0 &0 \end{bmatrix}\\ B= ...
0
votes
1answer
16 views

Sylvester's law of inertia for generic matrices.

By Sylvester's law of inertia, the positive and negative indices of a symmetric matrix $A$ are also the number of positive and negative eigenvalues of $A$. I was wondering if a similar result is known ...
0
votes
3answers
47 views

Inverse of partitioned matrices [closed]

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
2
votes
1answer
38 views

How to solve the convex combination problem of matrix?

Let $A \succeq B$ denote matrix $A-B$ is positive semidefinite, and here is the definition of redundant(all the matrix dimensions are $N\times N$ ): Given a set of matrix $\{B_i\}_{i=1}^{l}$, if ...
-3
votes
0answers
39 views

Vector spaces and nontrivial subspace. [closed]

Give an example of a subset of $\mathbb{R}^2$ that is a nontrivial subspace of $\mathbb{R}^2$? $\mathbb{R}^2$ as $\{(a, b) \mid a, b \in \mathbb{R}\}$
3
votes
0answers
55 views

Why we define the adjoint operator

Suppose in vector space $A: X\rightarrow Y$ is a linear map, the adjoint operator $A^{'}: Y^{'}\rightarrow X^{'}$ is defined as: $f(Ax)=(A^{'}f)(x)$. As I can understand, the adjoint operator just ...
2
votes
0answers
62 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
0
votes
2answers
32 views

In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space?

Say I have 3 distinct eigenvalues for a symmetric matrix. By the Spectral Theorem, the three eigenspaces are mutually orthogonal. But, if I just wanted to compute the first eigenspace, ...
1
vote
1answer
44 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
0
votes
0answers
35 views

Wedge product of maps: functorial vs. exterior algebra

Suppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{F}$. If $\varphi, \psi \in \hom(V,W)$, there are at least two interpretations of the symbol $\varphi \wedge \psi$: It is ...
0
votes
1answer
15 views

Stability of non-homogeneous and non-autonomous first-order difference equation

I am seeking to analyze the stability of steady points in a system of $n$ variables $x_1(t), ..., x_n(t)$. With discrete time $t$ the system is described by \begin{eqnarray*} x_i(t+1) = ...
1
vote
1answer
29 views

Identification between wedge product and its dual

Let $\mathbb{F}$ be a field, and let $(e_i)$ be the usual elementary basis of $\mathbb{F}^n$. Let $\varphi_{ij}: \mathbb{F}^n \wedge \mathbb{F}^n \to \mathbb{F}$ be such that $v \wedge w \mapsto ...
2
votes
2answers
46 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
2
votes
3answers
56 views

Producing lower bounds for $\text{trace}(A^2)$ for a positive semidefinite, symmetric matrix $A$

Are there any lower bounds on $\DeclareMathOperator{trace}{trace}$ \begin{align*} \trace(A^2), \end{align*} where $A$ is positive semi-definite and symmetric? I am aware of the inequality $$ ...
-1
votes
0answers
52 views

Find the solution to the following LPP by solving its dual. [closed]

Minimize : $ Z = 300X_1 + 110X_2$ Subject to : \begin{align*} 30X_1 + 5X_2 &\geq 6 \\ 20X_1 + 10X_2 &\geq 8 \\ X_1, X_2 &\geq 0 \end{align*}
0
votes
1answer
28 views

matrix multiplication manipulation

a,b $\in \mathbb{R^n}$ and C $\in \mathbb{R^{nxn}}$. I have $ab^TCab^TC$. I try to manipulate this multiplication into: $b^TCaab^TC$. I need help.
1
vote
2answers
25 views

Determining all scalars $a \in \mathbb{R}$ for which a matrixrepresentation is orthogonal?

Problem: Let $a \in \mathbb{R}$ and \begin{align*} T: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}: A \mapsto aA. \end{align*} Determine all $a \in \mathbb{R}$ for which the matrix of ...
7
votes
3answers
441 views

Rule for squaring arbitrary powers?

This is a really simple question, but I don't know how to phrase it well enough for Google. I'm going through a proof and don't understand how: $$ (q^{2^{n+1}})^2 = q^{2^{n+2}} $$ I thought it would ...
0
votes
2answers
27 views

Linear algebra: proving transformation matrix between orthogonal basis is unitary

The vector space $V$ is equipped with a Hermitian scalar product and an orthonormal basis $\{e_1,\ldots,e_n\}$. A second orthonormal basis $\{e_1',\ldots,e_n'\}$ is related to the first one by ...