Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

learn more… | top users | synonyms

0
votes
1answer
27 views

At most $n+1$ vectors, the angle between which $>\pi/2$.

In a $n$ dimensional Euclidean space $V$, there exists at most $n+1$ vectors, each pair has inner product $<0$. This is geometrically obvious in $3$ dimensions...But how can we prove it ...
1
vote
1answer
12 views

Finding pathline

I've been trying to find the pathline of a particle dropped in a steady flow defined by the following vector components: $$ u= \frac{-2x}{(x^2+y^2+1)^2} \hat i + \frac{-2y}{(x^2+y^2+1)^2}\hat j $$ in ...
0
votes
0answers
19 views

Prove that $V_i$ are $T$-invariant for $1\le i\le k$ and $V=\bigoplus_{i=1}^{k}V_i$

Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is ...
0
votes
0answers
8 views

Improvement of Minimum description length (MDL) estimate.

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
1
vote
1answer
37 views

Steinitz's Lemma - Removing

In the book that I am using, Linear Algebra Done Right, the proof for the Steinitz exchange lemma (which can be found here) left me unconvinced. The proof refers to the linear independence lemma. ...
0
votes
0answers
6 views

How to calculate a covariance matrix with given Canonical Correlation Analysis components and given variances/covariances for CCA components?

So given a covariance matrix, the Canonical Correlation Analysis (CCA) components can be computed along with the correlation between corresponding pairs of CCA components. What about the other way ...
0
votes
0answers
11 views

Space is direct sum of subspaces - propostion conditions giving me problems

In Sheldon Axler's "Linear Algebra Done Right" - 2$^{\textrm{nd}}$ Edition, on the section for Direct Sums, the following proposition is stated. Following this is the proof of this 'if and only ...
0
votes
1answer
19 views

Why the largest singular value of a megic matrix is its magic constant?

A magic matrix is a square matrix such that the sums of the elements of each row, each column and diagonal equal to a same number, the magic constant. As reported ...
4
votes
1answer
28 views

computation involving exterior $2$-form on $\mathbb{R}^n$

Let $$\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$$be an exterior $2$-form on $\mathbb{R}^n$, and $A, B \in \mathbb{R}^n$ are vectors$$A = (1, 1, 1, \dots, 1),\text{ }B = (-1, 1, -1, \dots, ...
2
votes
7answers
74 views

Prove that a matrix equals to its transpose

Let $A$ be a $(n\times n)$ matrix that satisfies: $AA^t=A^tA$ Let $B$ be a matrix such that: $B=2AA^t(A^t-A)$ Prove/disprove that: $B^t=B$ I started with: $$\begin{align} B &=2AA^t(A^t-A) \\ ...
0
votes
1answer
19 views

If the characteristic polynomials of $A$ and $B$ are equal, why are the corresponding coefficients of $\lambda^{n-1}$ equal?

Theorem: Similar matrices have the same trace. Proof: Let $A$ and $B$ be similar matrices. Then there is $P$, such that $B = P^{-1}AP$. Given that we have similar matrices then we also have ...
1
vote
0answers
15 views

Linear ODE and Fourier Series

Let $m,k_0,k$ be positive real numbers and $x_1$, $x_2$ be real-valued functions of time. Suppose we have following system of two coupled ODEs ( motivated by a coupled oscillator with two masses ...
2
votes
2answers
135 views

Is it possible that there isn't a linear span which precisely spans a vector space?

In the assignment I'm asked to decide whether given: $S = \Bigg \{ \begin{bmatrix}a &b \\ c &d\end{bmatrix} \in M_2(\mathbb{R}) \; | \; ad = 0 \Bigg \},\mathbb{F} = \mathbb{R}$. $S$ is a ...
5
votes
2answers
88 views

Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I enjoyed the subject a lot and I would like to know "what's next". In other words, I would like to know ...
3
votes
1answer
22 views

Find the area of a subset of $\mathbb{R}^3$ given by an implicit relation.

Let x, y, z be real numbers and let $A = \begin{bmatrix} 1&x&x^{2} \\ 1&y&y^{2} \\ 1&z&z^{2} \end{bmatrix} $ Let S be the subset of $\mathbf{R}^{3}$ given by $S = \{ ...
-5
votes
0answers
14 views

Is Frobenius norm of a gram matrix convex [on hold]

Suppose $X \in \mathbb{R}^{m \times n}$ and $S \in \mathbb{R}^{m \times m}$ Is the function $f(X) = ||XX^T - S||^2_{fro}$ convex w.r.t X ? Here, $S$ is a constant matrix. One can think of $XX^T$ as ...
0
votes
2answers
44 views

Intuitive transition from matrices to tensor-concept

I would like to know how to build intuition for the concept of a tensor using the following reasoning: If I conceive of a vector as an extension of the scalar concept, i.e. an $N \times 1$ "array of ...
1
vote
2answers
22 views

What row-operations allow this $\operatorname{Mat}_{2\times2} (\mathbb{R})$

$$ A = \begin{pmatrix} 1 & r \\ s & 1 \\ \end{pmatrix} \Rightarrow \begin{pmatrix} 1 & r \\ 0 & 1-s \cdot r \\ \end{pmatrix} = B \quad\quad r,s \in \mathbb{R} $$ Matrix B is ...
0
votes
1answer
15 views

Hermitian Matrix Inequality

If we have {$A_{ij}\}_{n*n}$ a Hermitian matrix. v=($v_1,v_2..v_n$), w=($w_1,w_2...w_n$) are two complex vectors. Then how can I show the inequality |$\sum_{i,j=1}^nA_{ij}v_i\overline{w_j}$|$\leq ...
1
vote
3answers
47 views

Derivative of a quadratic form

There is a Hermitian matrix $X$ and a complex vector $a$. I know that $a^HXa$ is a real scalar but derivative of $a^HXa$ with respect to $a$ is complex, $$\frac{\partial a^HXa}{\partial a}=Xa^*$$ Why ...
0
votes
1answer
18 views

simple moving average related to a mean

Am I right in this statement? Given a series of numeric values that represent measurements (y) over time (x), the closer a simple moving average is to the mean the less volatility in (y) ?
0
votes
0answers
24 views

Subspaces of vector space and their spans

$\def\sp{\operatorname{sp}}$ Given 2 subspaces of V, and $T\subseteq \sp(K)$ and $K\subseteq \sp(T)$ then $\sp(K)=\sp(T)$? If $ T\subseteq \sp(T)\subseteq \sp(K)$ and $K\subseteq ...
0
votes
2answers
35 views

Proof for pythagoras theorem

Let $f,g$ orthogonals to each other. $${\left\| {f + g} \right\|^2} = \left<f,f\right>+\left<g,f\right>+\left<f,g\right>+\left<g,g\right> = {\left\| f \right\|^2} + {\left\| g ...
2
votes
1answer
20 views

Find a hyperplane not intersecting $S$

I am struggling with the following problem: Let $K$ be an infinite field, $V$ an $n$-dimensional $K$-vector space, $S \subset V$ a finite subset with $0 \notin S$. Prove that there exists a subspace ...
2
votes
3answers
28 views

What ring-sum of vector spaces can possibly mean?

I'm given this test assignment, and I can't decipher what it says. Would you kindly help me? Here's the assignment itself: Let $U$ and $W$ be sub-spaces of the linear vector space $V$ s.t. $U ...
-6
votes
1answer
52 views

Inequality on matrix norm: $ \lVert A^n \rVert \leq \lVert A \rVert^n $ [on hold]

If $A$ is a $n \times n$ matrix and assume we have a matrix norm $\lVert \cdot \rVert$. In a proof I need the following property: $ \lVert A^n \rVert \leq \lVert A \rVert^n $. I don't know how to ...
1
vote
2answers
40 views

Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
0
votes
1answer
21 views

Proving $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}')$

If $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}') (\mathbf A'=transpose A) $
0
votes
1answer
24 views

How to find the velocity and accelaration in a 3d space with 6 degrees of freedom?

I have the following rigid body: I assume that the body is a symmetric cylinder.x,y,z are the axes of the reference frame resulting from a transformation involving three orthogonal rotations ...
1
vote
1answer
54 views

Book comparison, Linear Algebra

so next semester (Spring 2015) I'm taking a Linear Algebra class. I was wondering if anyone who's had this book "Linear Algebra and Its Applications, 4th Edition - by David C. Lay" can give me an ...
0
votes
1answer
20 views

Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
-2
votes
1answer
79 views

complex problem in linear algebra

Let $A$ be an $n$ by $n$ matrix. Let $D$ be an $n$ by $n$ diagonal matrix with distinct diagonal entries, and let $u$ be an $n$ by $1$ column vector with all non-zero entries. Let $Aq=\lambda q$ with ...
2
votes
2answers
25 views

Can a low-rank matrix set have nonempty interior?

The answer to this question may be super simple, but it is very not obvious to me. Consider the space $S^n$ of symmetric $n\times n$ matrices. Consider $T\subset S$ the set of rank $n-1$ matrices. ...
7
votes
1answer
109 views

Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
-4
votes
0answers
13 views

full row rank matrix and 2-norm solution

I am trying to solve this problem. Can you please give me an idea on how to solve this. $A$ is an $m$ by $n$ matrix with $m < n$ and with rank($A$)=$m$. Consider the system $Ax=b.$ 1. How to find ...
1
vote
1answer
68 views

Let $A$, $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = O$

Let $A$ and $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = 0$. Progress I know that $ABA=0 \implies A^2B=0$. Here ...
6
votes
2answers
79 views

What is the geometric interpretation of a vector squared?

I'm working through Introduction to Space Dynamics by William Tyrrell Thomson. I am having to do a lot of research to make it through even small parts, but I am unable to find information to make me ...
1
vote
0answers
31 views

Request for clarification about linear combinations

I need help understanding the basis of this statement in Axler's Linear Algebra Done Right, found on page 86 of the second edition: Because ($\vec{v_{1}}, \ldots, \vec{v_{n}}$) is a basis of $V$, we ...
3
votes
1answer
29 views

Change of basis matrix - part of a proof

I'm trying to understand a proof from Comprehensive Introduction to Linear Algebra (page 244) I can't really figure out what steps have been taken to get from eq. 1. to eq. 2. It's just ...
2
votes
1answer
17 views

Given $n$ linear functionals $f_k(x_1,\dotsc,x_n) = \sum_{j=1}^n (k-j)x_j$, what is the dimension of the subspace they annihilate?

Let $F$ be a subfield of the complex numbers. We define $n$ linear functionals on $F^n$ ($n \geq 2$) by $f_k(x_1, \dotsc, x_n) = \sum_{j=1}^n (k-j) x_j$, $1 \leq k \leq n$. What is the dimension of ...
1
vote
2answers
24 views

To determine Rank of Linear Transformation

Question is to find the rank of $T_1 $and $T_2$ Since the composition is bijective so rank of $T_1T_2 = m$. But how do I get the ranks of$ T_1 $and$ T_2 $from here? Thanks.
1
vote
1answer
36 views

Set of linear equations with coefficients - solution using matrices

I have a set of linear equations: \begin{matrix} ax_{1}& {}+bx_{2}& {}+x_{3}& & =0\\ cx_{1}& {}+dx_{2}& &{}-x_{4} & =0\\ & {}-ex_{2}& ...
3
votes
1answer
57 views

Help Understanding Proof from Linear Algebra Done Right

I'm doing a self-study of Axler's Linear Algebra Done Right, and am looking for some help understanding a step in the proof of Proposition 5.21, appearing on page 89 of the second edition. An ...
0
votes
0answers
37 views

A matrix transformation from R^4 to R^3 - linear algebra - how to find the image of a point

I'm trying to revise for an upcoming exam on linear algebra and have come across this question. I do not understand the line "the image of a point (x1, x2, x3, x4) can be computed from the defining ...
-1
votes
0answers
22 views

Algorithm for vector space transformation [on hold]

In my text book I've got an example which is as follows: Create an algorithm which calculates coordinates of a point after a space transformation took place. Transformations may be scaling or ...
2
votes
0answers
43 views

General form for the rotation of a function.

When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely. This was what I did to rotate $y=2x+1$ by ...
2
votes
3answers
40 views

Is $ x^n-y^n$ is a product of coprime factors?

In the expression: $x^n-y^n$, if $n>2$ and $x,y$ are relatively prime, are the factors $x-y$ and $ x^{n-1}+x^{n-2}y+.....$ always coprime? Why? Please exclude the cases where $x-y=\pm 1$ and $\pm ...
1
vote
1answer
58 views

How to solve this kind of problem?

I've just found the following problem: $\quad\quad$ $\quad\quad$ $\quad\,$ And I believe that it could be done with something in combinatorics, my feeling is that generating functions would ...
-3
votes
1answer
49 views

How do i find eigen vector

I need to find corresponding eign vector forthis problem Any hints for this .Thanks
1
vote
1answer
54 views

Vector Spaces: Tensor Product

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...