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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How can I find the co-ordinate of where a line bisects a circle?

I was looking to know if there was an equation that would allow me to calculate the co-ordinates of a point on the circumference of a circle where a line bisected it and the centre. My diagram should ...
4
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4answers
158 views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
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1answer
27 views

Show that every vector in the null space of $m \times n$ matrix $A$ is orthogonal to every vector in the row space of $A$.

How do I show? do I show this by using inner products? What should I define my inner product?
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0answers
24 views

Matrix polynomial [on hold]

Suppose: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a ...
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1answer
19 views

Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
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4answers
36 views

Equivalent definitions of an orthogonal matrix.

I wish to show that the following definitions of an $n \times n$ real matrix $Q$ are equivalent: $QQ^T=I$, $Qx\cdot Qx=x\cdot x$ for all $x\in \mathbb{R}^n$. I found it easy to show that ...
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0answers
13 views

E as an expectation of a quadratic form [on hold]

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
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1answer
28 views

Affinity of lorentz transformations

Lorentz transformations are often defined to be linear. But suppose instead we only consider transformations that preserve the spacetime interval. Is it possible to prove that those transformations ...
4
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1answer
24 views

Determinant inequality and positive definite matrix

Let $B$ and $C$ be $n\times n$ hermitian matrices, with $B$ positive definite and $C$ positive semi-definite. (1) Show that $B+C$ is positive definite (2) Show that $\det(B)\leq \det(B+C)$. What ...
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0answers
12 views

Let T be the bounded operator and T* be the adjoint operator of T.Show the following. [on hold]

Let T be the bounded operator and T* be the adjoint operator of T. Show the following. 1.||TT||=||T|| 2.||TT||=||TT*||=||T||^2 3.(T+S)=T+S* 4.(αT)=α ̄T (α∈C) 5.(TS)=S T* 6.(T* )* =T
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16 views

System of Equations for 3-digit number [on hold]

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
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2answers
46 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
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1answer
24 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 ...
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1answer
48 views

Verify the following assertion: [on hold]

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.
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0answers
59 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 ...
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0answers
20 views

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

things I need 1. a visual recpinatation 2. A explanation on how to solve it and 3. a problem to do
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2answers
47 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 ...
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0answers
21 views

Maximization of sum of convex 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) is concave ...
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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 ...
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0answers
27 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.
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26 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
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1answer
54 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
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1answer
13 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*} ...
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8answers
144 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.
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0answers
16 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
24 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
87 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 $b$, ...
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0answers
27 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 ...
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4answers
60 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 ...
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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
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2answers
29 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
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2answers
38 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
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4answers
47 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 ...
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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 ...
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2answers
23 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
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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
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1answer
31 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, ...
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0answers
19 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
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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. $CA=B$
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1answer
29 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 ...
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1answer
17 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 ...
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1answer
49 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= ...
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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
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3answers
40 views

Inverse of partitioned matrices [on hold]

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
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1answer
31 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 ...
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0answers
32 views

Vector spaces and nontrivial subspace. [on hold]

Give an example of a subset of $\mathbb{R}^2$ that is a nontrivial subspace of $\mathbb{R}^2$? $\mathbb{R}^2$ as $\{(a, b) \mid a, b \in \mathbb{R}\}$
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0answers
52 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
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0answers
57 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 $K$, ...
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2answers
23 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
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1answer
33 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, ...