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 ...

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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) \\ ...
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Why are the coefficients of λn-1 in ø(A,λ) = coefficients of λn-1 in ø(B,λ)when proving that similar matrices have the same trace?

Theorem: Similar Matrices have the same trace. Proof: Let A and B be similar matrices. Then Ǝ P, s.t ...
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10 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
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2answers
54 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 ...
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61 views

Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra (3rd 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 what are the ...
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20 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 = \{ ...
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Is Frobenius norm of a gram matrix convex [on hold]

Is the function $f(X) = ||XX^T - S||^2_{fro}$ convex wrt X ? Here, S is a constant matrix. One can think of $XX^T$ as the Gramian matrix.
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38 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 ...
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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 ...
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1answer
14 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 ...
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39 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 ...
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1answer
15 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) ?
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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 ...
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2answers
33 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
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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 ...
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3answers
27 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 ...
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1answer
48 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 ...
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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 ...
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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) $
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1answer
23 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 ...
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1answer
50 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 ...
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1answer
19 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) ...
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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
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1answer
95 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: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
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0answers
12 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 ...
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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 ...
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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
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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
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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 ...
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2answers
23 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.
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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
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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 ...
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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 ...
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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
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0answers
42 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
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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 ...
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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 ...
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1answer
49 views

How do i find eigen vector

I need to find corresponding eign vector forthis problem Any hints for this .Thanks
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51 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 ...
2
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3answers
151 views

Linear dependency of nilpotent matrices

I would like to prove that four $2\times 2$ nilpotent matrices are always linearly dependent, using the Cayley-Hamilton theorem or the minimal polynomial in some way. I think I have proved the ...
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7answers
112 views

Given matrix P such that $P^{102 } =0 $ , to show that $P^{2} = 0$.

P is given to be a 2×2 matrix such that $P^{102} = 0$. How to show that $P^{2} =0 $?
2
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1answer
26 views

If A and A' are approximately the same, are their principal components/SVD very close?

If we have that two matrices $A\approx A'$ within some guaranteed error bound for each term, and $A=U\Sigma V$ is the singular value decomposition for $A$, and $A'=U'\Sigma' V'$ is the SVD for $A'$, ...
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3answers
17 views

To find the two dimensional subspace of $R^{3}$

I am stuck with this question .Kindly help me to get through this Option A is of 1 dimension so it cannot be answer but all other options are looking fine to me , What i am missing ? THANKS
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2answers
47 views

Using inverse of matrix A as approximate inverse of matrix that is very close to A

Say we have two matrices, $A$ and $A'$ so that $A\approx A'$, and we have the inverse of $A$, $B$, where $AB=I$, and the inverse of $A'$ where $A'B'=I$. If we have some guarantee about how big any ...
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0answers
24 views

Proving matrix similarity for given matrices

Let $\mathbb{F}$ be a field, and let $A=(a_{ij})_{i,j=1}^n$ and $B=(b_{ij})_{i,j=1}^n$ be matrices in $M_n(\mathbb{F})$ such that: a. $b_{ij}=0 \iff a_{ij}=0$ $ \forall 1 \leq i,j \leq n$ b. ...
0
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1answer
17 views

Null/Col/Row space be a line\plane through the origin?

For a $4\times3$ matrix can the nullspace, the column space and row space all be lines through the origin? For a $2\times4$ matrix can the nullspace, the column space and row space all be planes ...
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1answer
23 views

Two square matrices with the same minimial polynomial are similar for $n=5$ or $n=6$

Let $\mathbb{F}$ be a field, $\lambda \in \mathbb{F}$ and $A,B \in M_n(\mathbb{F})$ such that $m_A(x)=m_B(x)=(x-\lambda)^k$ and such that the geometric multiplicity of $\lambda$ in $A$ equals to the ...
2
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0answers
38 views

How can i find column of matrix corresponds to row of matrix's inverse

let $Y=X\beta$ be an equation of matrix and let $X$ be an invertible $n\times n$ matrix, $Y$ be $n \times 1$ matrix, $\beta$ be $n \times 1$ matrix. $$\begin{bmatrix} y_1 \\ y_2 \\y_3 ...