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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4
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1answer
12 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the $2 \times 2$ matrix ring $M_2(\mathbb{C})$. let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional ...
0
votes
0answers
7 views

How to make use of symmetric of sparse matrix to solve this kind of problem?

I have the following matrix to be solved: $$\left\{ \matrix{ {a_{11}}{x_1} + {a_{12}}{x_2} + \cdots + {a_{1n}}{x_n} = {y_1} \hfill \cr {a_{21}}{x_1} + {a_{22}}{x_2} + \cdots + {a_{2n}}{x_n} = ...
0
votes
1answer
10 views

Dependence of product of matrix and a vector, on the rank of a Matrix

What is the significance of the rank of a matrix, say $A$, when I am multiplying a vector, say $x$, by $A$? In other words, let $x$ be a column vector of suitable dimension and let $rank (A)=m$. What ...
2
votes
2answers
34 views

$\frac{1}{{1 + \left| {\left\| A \right\|} \right|}} \le \left| {\left\| {{{(I - A)}^{ - 1}}} \right\|} \right|$

Let a matrix norm $\left| {\left\| . \right\|} \right|$ have the property that $\left| {\left\| I \right\|} \right| = 1$ and $\left| {\left\| A \right\|} \right| < 1$. Why does the following ...
0
votes
0answers
26 views

How to find one matrix, which is subject to $B^3 = A$. How much is such matrices?

Here I have a problem with row echelon form. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
0
votes
0answers
12 views

Invertibility for a matrix that I don't know

I would like to know why $(e^{-At}-I)^{-1}$ is invertible when matrix A is Hurwitz.
1
vote
0answers
8 views

How to modify Tikhonov regularization?

Consider a linear map $f: X \rightarrow Y$ and let $F$ is a matrix of $f$ and $b$ is one element of $Y$. Our goal is to obtain the element of $X$ corresponding to $b$. In ideal case we can get the ...
0
votes
1answer
22 views

Show that the matrix is a symmetric matrix

Let $T:V\to V$ be a symmetric linear map i.e $\langle Tx,y\rangle =\langle x,Ty\rangle $ .$V$ is a finite dimensional inner product space If $\{e_i:1\leq i\leq n\}$ is an orthonormal basis of $V$ ...
3
votes
3answers
18 views

Determinant of symplectic matrix

A $2n \times 2n$ matrix $S$ is symplectic, if $SJ_{2n}S^T=J_{2n}$ where \begin{equation} J_{2n} = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}. \end{equation} My question is, how to ...
0
votes
6answers
43 views

How to prove there's a vector $z \in \mathbb{R}^4$ orthogonal to two linearly independent vectors $x,y \in \mathbb{R}^4$?

Let $x, y \in \mathbb{R}^4$ with $\{x, y\}$ being linearly independent. Prove that there exists a non-zero vector $z$ that is orthogonal to both $x$ and $y$. Any hints on what to do after the ...
1
vote
2answers
31 views

If Q is an orthogonal matrix, does it follow that $QDQ^T = Q^TDQ$?

Say A is a real, $n \times n$ symmetric matrix. Then it is orthogonally diagonalisable, with $A = QDQ^T = QDQ^{-1}$. Let's say we do not know that Q is symmetric (at first) - does the above hold?
2
votes
3answers
33 views

Can systems of 3 linear equations with 3 unknowns have more than one solution?

In each part,determine whether the given vector is a solution of the linear system \begin{align} 2x-4y-z&=1\\ x-3y+z&=1\\ 3x-5y-3z&=1 \end{align} (a) $(3,1,1)$ (b) $(3,-1,1)$ (c) ...
-4
votes
0answers
25 views

Is this always true about multiplication of matrices [on hold]

Let $\mathbf{A},\mathbf{B}$ be square matrices of size $n\times n$. Assume that $\mathbf{A}$ is symmetric and $\mathbf{B}$ is non-singular. Will $\mathbf{Y} = \mathbf{BAB}^T$ be always a symmetric (or ...
0
votes
0answers
22 views

Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
1
vote
1answer
13 views

Invertibility of Product implies invertibility of factors

Say $C=AB$ where $A,B,C$ are all $n\times n$ matrices. It's easy to show that if $A$ and $B$ are invertible then $C$ is invertible --> $C^{-1}=B^{-1}A^{-1}$. Does the converse hold? That is, if $C$ ...
0
votes
1answer
37 views

$\left\| {\left| {BA - I} \right|} \right\| < 1$ $ \Rightarrow $ $A$ and $B$ are both nonsingular

Let $A,B \in {M_n}$ satisfy the inequality $\left\| {\left| {BA - I} \right|} \right\| < 1$ and $\left\| {\left| . \right|} \right\|$ be a matrix norm on ${M_n}$.Why do $A$ and $B$ are both ...
0
votes
1answer
19 views

reordering the indices of a matrix

Let $A$ be an $n \times n$ matrix of rank r. Then by reordering the indices if necessary we can bring the matrix in the form $(\frac{A_1}{A_2})$ where $A_1$ is an $r \times n$ matrix, $A_2$ is an $n-r ...
0
votes
3answers
38 views

Let {v1, v2} be a basis for a subspace S of R 3 . If B = {w1, w2, w3} is a set of vectors in S, then B cannot be linearly independent.

Let $\{v_1, v_2\}$ be a basis for a subspace $S$ of $\Bbb R^3$ . If $\mathcal B = \{w_1, w_2, w_3\}$ is a set of vectors in $S$, then $\mathcal B$ cannot be linearly independent. I'm not sure how ...
1
vote
3answers
26 views

Use Vectors To Show Three Vertices Belong to a Right Triangle

The Full Question Theorems Used This is what I call theorem 1: My Work This problem has two major steps as far as I can see. First, I must show that these are points of a triangle(not ...
4
votes
1answer
33 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
0
votes
1answer
22 views

Express a second-order cone (SOC) inequality as a linear matrix inequality (LMI)

For $y \in \mathbf{R}^n$ and $t \in \mathbf{R}$, show that: $$||y||_2 \leq t ~~\iff~~ F(y) \succeq 0$$ Where $\text{I}$ is the $n \times n$ identity matrix, and $$F(y) = \begin{pmatrix} t ...
0
votes
2answers
42 views

Show that $\mathcal{B}$ is a Basis for $V$

If $V= \{p(x) \in \mathbb{R}_3[x] : p(-1)=p(1)=0\}$, show that $\mathcal{B} = \{ 1 - x^2, x - x^3\}$ is a basis for $V$. Note: $\mathbb{R}_3[x]$ denotes polynomials with real coefficients of degree ...
-3
votes
0answers
20 views

Linear Algebra: Guidance on a Eigenvalue/Eigenbasis problem, please?

Here's the problem, but I only need some help with part C: http://i.imgur.com/UwRBGIO.png This is the information and answers from the back of the book: http://i.imgur.com/BFs2z2s.png I understand ...
1
vote
1answer
28 views

Area Between Intersecting Lines - Elegant Solution?

I am running simulations, and the output will be a line y = mx+b. I am interested in the area below the line between x=0 and x=1. I am only interested in the area that is below the diagonal y = x. I ...
2
votes
1answer
33 views

Is my algorithm correct? (Polar decomposition)

I cant seem to find my mistake. Consider this matrix $T = $\begin{bmatrix} 2 & 1 & 1 \\[0.3em] -1 & 2 & 0 \\[0.3em] 0 & 1 & -1 \end{bmatrix} I need ...
2
votes
3answers
24 views

Gradient of a line

The line L is a reflection of the line $2y + 3x =9$ in the $y-$ axis (I had to draw the graph on the grid previously) Find gradient of the line L How would I go about solving this?
2
votes
1answer
21 views

Understanding the Replacement Theorem (Exchange Theorem)

I'm learning about Basis and Spans and now that's I've figured out what these are, I'm trying to understand the Replacement Theorem(also called the Exchange Theorem). The definition goes like this: ...
0
votes
0answers
8 views

Maximizing Autoencoder Hidden Unit Function

Given \begin{align} a = f\left(\sum_{j=1}^{100} W_j x_j \right). \end{align} where $f$ is the sigmoid function, $W$ and $x$ are $100 \times 1$ matrices with the constrain \begin{align} ||x||^2 = ...
1
vote
2answers
21 views

Show that the set is a basis for $S$.

Consider the subspace $S$ in $\Bbb R^3$, $S=\{(a,b,c)\mid a+b=c\}$. Show that the set $B= \{(1,0,1),(1,2,3)\}$ is a basis for $S$. I've started to set up a matrix, ...
1
vote
0answers
42 views

Can a matrix be similar to more than one matrix?

I have a little query about similar matrices I've been struggling with. Suppose I have a 5x5 diagonal matrix A with 5 distinct eigenvalues as entries in the main diagonal. The question is, to how ...
0
votes
0answers
27 views

Reflection matrix and algebraic multiplicity

Let $Q\in\mathbb{M}_4(\mathbb{R})$ a reflection matrix onto $R(A)$ subspace, where $A\in\mathbb{M}_{4\times 3}(\mathbb{R})$ is defined by ...
2
votes
4answers
29 views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
2
votes
1answer
24 views

Show that the subset $S$ in $\mathbb{R}_3$ is a subspace.

Show that the subset $S$ in $\mathbb{R}_3$ defined by $S=\{(a,b,c) \in \mathbb{R}_3 \text{ such that } a+b=c \}$ is a subspace. I'm having trouble adapting the definition of subspace with the part ...
3
votes
1answer
37 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
1
vote
1answer
13 views

Is it possible to find a vector that is orthogonal to this set?

I have a set of four vectors in $\mathbb{R}^4$: $\{ \vec v_1, \vec v_2, \vec v_3, \vec v_4 \}$ The first three are linearly independent, but $ \vec v_4 $ is a linear combination of the others. Is it ...
1
vote
2answers
24 views

demonstrate that $v_3 \perp (v_1-v_2)$

$\bar{v}_1 \perp (\bar{v}_2-\bar{v}_3)$ and $\bar{v}_2 \perp (\bar{v}_3-\bar{v}_1)$ therefore $\bar{v}_3 \perp (\bar{v}_1-\bar{v}_2)$ By applying the dot product to $\bar{v}_1$ and ...
1
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1answer
40 views
1
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3answers
43 views

How are these expressions equivalent?

I saw that $${y^2\over y^2+1} = 1 - \frac1{y^2+1}$$ but I can't see how, wolfram alpha agrees but I'm still not seeing it.
1
vote
0answers
15 views

Polar decomposition varient

I have a factorisation to do, and I think that a varient of Polar decomposition will give me what I need, although I'm not sure of the exact form. I have \begin{equation*} \mathbf{y} = ...
0
votes
1answer
103 views

Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0

Given a matrix $A$ find a two dimensional subspace $V \subset\mathbb{R}^4$ for which $\forall x \in V : x^TAx=0$ $$A = \begin{pmatrix}1&2&0&1\\ 2&3&1&1\\ 0&1&0&1\\ ...
1
vote
2answers
41 views

Rotation matrix

I'm finding different results for the 3D rotation matrix in the XY plane from different sources and I was hoping for someone to help clarify. In my "applications of vector calculus" book, the matrix ...
7
votes
1answer
77 views

$A$ is diagonalizable if $A^8+A^2=I$

Given a matrix $A\in M_{n}(\mathbb{C})$ such that $A^8+A^2=I$, prove that $A$ is diagonalizable. So let $p(x)=x^8+x^2-1$ and we know that $p(A)=0$. The next step would be to show that the algebric ...
0
votes
2answers
22 views

methods of constructing a matrix from its null space span

I have a matrix of size $4\times3$ and its null-space span is $\{(1,2,3), (2,5,7)\}$. How can I find the original matrix? It is not obvious from the span which vectors are free.
0
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2answers
34 views

I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $$\begin{align} T\left(-2 x^2\right) &= 3 x^2 + 3 x \\ T(0.5 x + 4) &= -2 x^2 - 2 x - 3 \\ T\left(2 x^2 - 1\right) ...
0
votes
3answers
57 views

Self-Study Linear Algebra book for a complete understanding

I recently took an introductory class on linear algebra (covered solving linear systems, determinants, eigenvectors, diagonalization, some vector spaces, basis and combinations, transformations etc.) ...
2
votes
2answers
25 views

Algebric and geometric multiplicity and the way it affects the matrix

Given a matrix $A$. Suppose $A$ has $\lambda_1,\dots,\lambda_n$ eigenvalues each with $g_i$ geometric multiplicity and $r_1,\dots,r_n$ algebric multiplicity, $g_i\leq r_i$. Given this information ...
4
votes
3answers
67 views

Why does $\frac{1}{{\left\| {\left| {{A^{ - 1}}} \right|} \right\|}} \le \left\| {\left| B \right|} \right\|$?

Let $A,B \in {M_n}$ suppose that the following statements are true: $A$ is nonsingular, $A+B$ is singular, $\left\| {\left| . \right|} \right\|$ is matrix norm. Why is it true that: ...
0
votes
0answers
16 views

Find a generator for vectorial subspace

S = {$(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0$} $c+d-i(2ia)=0$ $c+d+2a=0$ $c=-d-2a$ $(a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1)$ Is this solution correct?
-1
votes
1answer
25 views

I have to show $a_{nn} \neq 0$. [on hold]

Let $D$ be an algebraic division ring with center $F$. $A,B$ are upper triangular matrices in $M_n(D)$. let $ A=\begin{pmatrix} a_{11}&a_{12} & \ldots &a_{1n}\\ 0 & a_{22} & ...
0
votes
1answer
14 views

To find nullity of a surjective linear mapping

Let $T:U \to V$ be a surjective linear mapping and $dim(U)=6$,$dim(V)=3$.Then a) $dim(ker$ $T)$ is greater than $4 $ b) $dim(ker$ $T)$$ = 4 $ c) $dim(ker$ $T)$ is greater than $3 $ d) $dim(ker$ ...