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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1answer
10 views

Clustering of vectors via inner product relationship

This might be an odd question, but suppose I have a lot of vectors $a_i\in\mathbb{R}^{3}$ (not necessarily unit) and for some unit vector $u\in\mathbb{R}^{3}$ I find $$ \sum_{i=1}^m ...
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1answer
10 views

How to find the span for a linear transformation?

I'm learning Linear Transformations and I understand what is a linear transformation. Now I'm trying to look at an example question and I'm not really sure how the span is found. The question goes as ...
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4answers
33 views

Prove that $\|v \|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$

Suppose $(e_1,\cdots, e_m)$ is an orthonormal basis in $V$. Let $v \in V$ . Prove that $\|v\|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$ Let $v\in V$ and ...
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0answers
11 views

What is wrong with my solution to this problem?

The base $ABCD$ of the figure has area $9$. The point $M$ divides the segment $AB$ on ratio $2$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that ...
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0answers
4 views

How many binary vectors of weight 3 can you have before their span contains one of weight 2?

In other words, I am looking for the smallest $k$ for which the following is always true: Let $v_i \in \mathbb{F}_2^n$ for $i = 1\ldots k$ be distinct vectors of Hamming weight 3, that is, vectors ...
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0answers
11 views

Behavior of MGF of Quadratic Combination of Dependent Multivariate Gaussians

Sorry if the formatting is poor, this is my first time asking a question. I'm investigating how squared gaussians behave, using the techniques provided here, which are giving me inconsistent results. ...
3
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1answer
30 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A_1,A_2 \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ ...
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0answers
23 views

Algebra Word Problem Help 1337 [duplicate]

Jamal borrowed $\$7,000$ to buy a used car. He borrowed some of the money from a bank that charged $7.8\%$ simple interest and the rest from a friend who charged $9\%$ simple interest. If the total ...
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2answers
27 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
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0answers
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Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
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1answer
20 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
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0answers
37 views

Algebra Word Problem 1337 [on hold]

Jamal borrowed $\$7,000$ to buy a used car. He borrowed some of the money from a bank that charged $7.8\%$ simple interest and the rest from a friend who charged $9\%$ simple interest. If the total ...
1
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3answers
38 views

Uniqueness of basis vectors

Say I have 2 vectors $v_1$ and $v_2$ as basis of a subspace. Then is it true that $kv_1$ and $mv_2$ where $k$ and $m$ are real numbers, are also basis for that subspace?
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0answers
16 views

How would we generate a basis of sigmoidal functions?

I am trying to figure out how to generate a basis of sigmoidal functions. My issue is thus: there are several possible generating functions for a sigmoid (logistic curves, error functions, arctangent, ...
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1answer
24 views

Let A and B be n*n matrices such that trace(A)<0<trace(B).

Let A and B $n\times n$ such that trace(A)$\lt0\lt$trace(B). Then, $f(t)=1-det(e^{tA+(1-t)B})$ has 1) infinitely many zeros in $0\lt t\lt1$ 2) at least one zero in $\Bbb R$ 3) no zeros 4) either ...
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3answers
44 views

Are $\cos$ and $\sin$ linearly dependent in $[- π , π]$? [on hold]

Are $\cos$ and $\sin$ linearly dependent on $[- π , π]$ If true, demonstrate; if False show a counterexample.
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1answer
26 views

I have to show center of $M_n(H)$ is $\mathbb{R}I$

Let $H$ be a real quaternion ring. I have to show that the center of $M_n(H)$ is $\mathbb{R}I$. Can anyone help?
3
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3answers
195 views

How do I restrict k to ensure my matrix has exactly 3 distinct eigenvalues?

$$A=\begin{bmatrix}-1&-1&0\\-12&3&-1\\k&0&0\end{bmatrix}$$ How do I restrict $k$ to ensure that my matrix has 3 distinct real eigenvalues? I tried going about it the long way ...
2
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0answers
20 views

Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
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2answers
45 views

Distance between points

Suppose I have two matrices each containing coordinates of $m$ and $n$ points in 2 D. Is there an easy way using linear algebra to calculate the euclidean distance between all points (i.e., the ...
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0answers
40 views

A question in matrix norm [on hold]

Let $I,A \in {M_n}$ and suppose $\left| {\left\| . \right\|} \right|$ be a matrix norm $\left| {\left\| I \right\|} \right| \ge 1$ and $\left| {\left\| A \right\|} \right| < 1$($I$ is identity ...
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1answer
49 views

Determinant proof using its properties

Prove without expanding: \begin{equation} \begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = ...
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2answers
53 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 ...
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0answers
20 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} = ...
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votes
1answer
17 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 ...
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3answers
79 views

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

Let a matrix norm $ {\left\| . \right\|}$ have the property that $ {\left\| I \right\|} = 1$ and $ {\left\| A \right\|} < 1$. Why does the following inequality hold? $$\frac{1}{{1 + \left\| A ...
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0answers
43 views

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

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

Invertibility for a matrix that I don't know [on hold]

I would like to know why $(e^{-At}-I)^{-1}$ is invertible when matrix A is Hurwitz.
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1answer
26 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
23 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 ...
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6answers
68 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
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2answers
35 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?
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3answers
81 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) ...
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0answers
25 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
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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 $C$ ...
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1answer
41 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
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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 ...
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3answers
45 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 ...
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3answers
29 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
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1answer
41 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 ...
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1answer
32 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 ...
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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 ...
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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 ...
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1answer
34 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
36 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
27 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
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1answer
25 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
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0answers
12 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
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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
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0answers
52 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 ...