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 ...

learn more… | top users | synonyms

0
votes
1answer
37 views

Basis for set of polynomials

I just did a problem where I think I am able to draw the following conlusion The set of all polynomials with complex coeff of degree $\le n$ has the same basis as The set of all ...
0
votes
1answer
53 views

calculate svd - example with roots

Do an Singular Value Decomposition of $$ \begin{bmatrix} 0 & \sqrt{2} & 0 & \sqrt{2} \\ \sqrt{2} & 0 & \sqrt{2} & 0 \end{bmatrix}$$ I have tried to find it following the ...
1
vote
1answer
71 views

Project $x$ orthogonally onto $v$: is the projection $p$ the average of $x$?

I have made a simple Freemat script as the following: ...
1
vote
3answers
37 views

A and B is similar ⇒ $A^T$ is similar to $B^T$.

Prove that, if A and B is similar, then $A^T$ is similar to $B^T$. Attempt: I try to use the defn of similarity, and then the rule to transpose in different order, but get nowhere. This should be ...
2
votes
4answers
163 views

Product of reflections is a rotation, by elementary vector methods

Let $\mathbf{u}$ and $\mathbf{v}$ be two 3D unit vectors. The transform that performs reflection in the plane normal to $\mathbf{u}$ is given by $$ T_{\mathbf{u}}(\mathbf{x}) = \mathbf{x} - ...
1
vote
1answer
29 views

Find the value of $n$ where

Let $f(x+y)=f(x)f(y)$ where $x,y$ are natural no.s, and $f(1)=3$ Then find $n$ if $\sum_1^nf(n)=120$
5
votes
1answer
77 views

Construct a rational matrix $A$ s.t. $A^m = I$

Let $K$ be a field of either $\mathbb{C}$, $\mathbb{R}$ or $\mathbb{Q}$, Let $V$ be a $n$ dimensional vector space over $K$. I want to construct a matrix $A \in GL(V)$ s.t. $A^m = I$ for some $m$ and ...
0
votes
2answers
59 views

asking about orthogonal matrix

1) How to proof this as orthogonal matrix U preserve angles: $$<Ux,Uy> = <x,y>$$ for any x and y. 2) How to prove that eigenvalue have unit absolute value?
0
votes
1answer
84 views

Linear Algebra quick question over isomorphism

can someone provide me an example of two isomorphic subspaces of r2 that are not identical? I am just curious since I can only find ones that are identical
0
votes
2answers
343 views

Distance problem - aptitude

Tim and Elan are 90 km from each other.they start to move each other simultanously tim at speed 10 and elan 5 kmph. If every hour they double their speed what is the distance that Tim will pass until ...
0
votes
1answer
46 views

Linear Algebra 2 Quick questions regarding my understanding of isomorphism

I know the definition of isomorphism but can you provide me two isomorphic subspaces of $\mathbb R^2$ that are not identical, and an example of a set that spans a subspace of $\mathbb R^3$ but is not ...
1
vote
3answers
63 views

Linear Algebra Vector Space matrix help

Let $M_{2\times2}$ be a vector space of all $2\times2$ matrices. If the transformation from $M_{2\times2}$ to $M_{2\times2}$ is $t(A)=A+A^T$ and $A$ is a $2\times2$ matrix with the top row $a,b$ and ...
2
votes
0answers
59 views

Cauchy-Schwarz on a Euclidian Space

I was thinking about this proof of the cauchy-schwarz inequality, I wanna show that $$|\langle u,v\rangle|\leq|u||v|$$. We know that, $$|\langle u,v\rangle| = ||u||v|\cos{\theta}|$$ where $\theta$ ...
3
votes
2answers
107 views

A function that looks like determinant

Let $A$ be the $n\times n$ matrix $(a_{ij})$. By Laplace formula, the cofactor expansion along the $j$th row is $$\det(A)=\sum_{j=1}^n (-1)^{i+j}a_{ij}M_{ij}.$$ I'm studying the function ...
0
votes
1answer
333 views

Linear Algebra Questions over vector Spaces

I have 3 questions and I am wondering whether or not the following are always true or never true. 1.If the null space of 5x4matrix is 2D, then the column space of the matrix can be isomorphic to a ...
3
votes
2answers
68 views

How can I represent an N dimensional line?

How can I represent a straight line (between two points) in a N-dimensional space?
0
votes
3answers
37 views

Proof of normalvector on a plane

I found that, for the plane with linear equation: Ax + By + Cz = 0, that the vector a with coordinates: (A, B, C), is a normal vector on that plane. Where does that come from? And can someone provide ...
0
votes
1answer
287 views

Linear Algebra Vector True and False Questions

I have a few true and false questions. I have explanations for them could someone please check them over? $R^3$ contains two disjoint subspaces. I think this is true for example {1,2,3} and {4,5,6} ...
1
vote
0answers
74 views

Would someone be knowledgeable enough to understand Wikipedia's proof of Schur Product Theorem?

It basically states that the Hadamard product of two semidefinite matrices are in fact semi definite. The proof from Wikipedia: ==== Proof of positivity ==== Let $M = \sum \mu_i m_i m_i^T$ and $N = ...
2
votes
1answer
40 views

Relationship betwen image components and SVD?

Let's say I have an image representing a sampled function. It just so happens that I know this function can be represented as a sum of individual outer products along with some noise. So I might ...
2
votes
2answers
59 views

If $T$ is diagonzalizable, should the following be a direct sum?

In his notes, my instructor began with the definition that $T \in L(V)$ is diagonalizable $\leftrightarrow$ $V$ has a basis each vector of which is an eigenvector of $T$. Let $\lambda_1, \lambda_2 ...
0
votes
0answers
80 views

problem computing block inverse of factorized matix

I'm running into trouble when verifying the inverse of the following block matrix using the Schur complement: My matrix is given by: $$K = USU^T,$$ where U are its eigenvectors and S the ...
4
votes
0answers
68 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
0
votes
1answer
60 views

Can I rotate divergence away? / Can get divergence from a rotation?

Let $v(x,y)$ be a two-dimensional vector field and let $R(x,y, \theta)$ be the two-dimensional rotation matrix which rotations a vector field around $(x,y)$ an angle $\theta$. The following two ...
2
votes
1answer
173 views

Partial derivative of double sum

Part of a homework question asks us to find the partial derivative $\frac{\partial C}{\partial x_i}$ of $$C(x)=(Ax−r)^2 =\sum_{i=1}^n\left(\sum_{j=1}^m A_{ij}x_j - r_i\right)^2$$ where $A$ is an ...
0
votes
2answers
80 views

How do I calculate the new x y coordinate for a rectangle when centering it within a rectangle?

I need to center a rectangle inside another rectangle. I know the width and height of the parent rectangle, and I know the width and height of the child rectangle that needs to be centered. I need ...
1
vote
0answers
37 views

Direct limit, density and norm

Let $E$ be a Banach space, $A_n$ be an increasing sequence of finite dimensional subspaces of $E$, $B_n$ be an increasing sequence of subspaces of $A_n$ and let $C_n = A_n/B_n$. Assume that the ...
0
votes
1answer
30 views

Minor problem in an exercise concerning linear subspace

This is the first exercise in Peter Lax's "Functional Analysis". He claimed that The sum of any collection of linear subspaces $\{S_i\}_{i\in I}$ is a linear subspace $S$. ($I$ is the index set) ...
2
votes
3answers
137 views

Prove $\det(A)=\det(A^T)$ detail

I want to prove that $$\det(A)=\det(A^T)$$ and the one step I don't understand (the problem is guiding you thought it is to prove $$P^T_{\sigma} = P_{\sigma^{-1}}$$ where P is a permutation matrix. ...
1
vote
0answers
27 views

Find maximum rank vector

I have a $n\times n$ matrix with coefficients from a finite field $F$. For any vector $v$ of $F^n$, I consider the sequence: $v_1=v$ and $v_{n+1}=M.v_n$ The rank of $v$ is the rank of the sequence ...
5
votes
2answers
79 views

Randomly generate an matrix $A$ s.t. $A^m = I$

Fixed $n$, I want to randomly generate a $n \times n$ real matrix $A$ from the set: $\{A \in \mathcal{M}_{n \times n}(\mathbb{R}): \exists m \in \mathbb{N} \mbox{ s.t. } A^m = I\}$ I think I should ...
1
vote
2answers
1k views

Easiest way to solve system of linear equations involving singular matrix

I am trying to balance an unbalanced chemical equation by using setting up a system of linear equations to solve for the stoichiometric coefficients in the chemical equation. After setting up a ...
0
votes
1answer
29 views

If $d$ is a singular value of an operator $T$, then is $d^2$ a singular value of $T^2$?

I'm trying to prove/disprove a homework problem that is the title question. I'm not looking for an explicit answer, just some direction. So, I've been reading Axler's book Linear Algebra Done Right ...
1
vote
3answers
76 views

Extending basis of a subspace

(From Gilbert Strang's Linear Algebra(4e). Section 3.5 . Problem 22) Question: Suppose S is a 5-dimensional subspace of $R^6$. True or false. a) Every basis for S can be extended to a basis for ...
5
votes
4answers
122 views

Show that $Q_8$ can't be embedded in $M_{2 \times 2}(\mathbb{R})$ as a group.

So, suppose that we're working in a field $F$. Consider the ring $M_{n \times n} (F)$ which is the set all $n \times n$ matrices with entries in $F$. Is it possible to determine whether a matrix ...
1
vote
1answer
227 views

Proof of Schur Product Theorem

Does anyone know how I can find a proof using operators of Schur's Product Theorem? Most proofs I have seen are very terse. Is there a way to prove it with operators and their matrices? Thanks!
1
vote
5answers
235 views

Find basis so Transformation Matrix will be diagonal

$e_1,e_2$ will be basis for $V$. $W$ has a basis $\{e_1+ ae_2,2e_1+be_2\}$. Choose an $a,b$ s.t. that the basis for $W$ will have a transformation matrix $T$ will be in diagonal form. $T(e_1) = ...
0
votes
1answer
38 views

Proof of Convergence of K-means

$T(e_1) = e_1+5e_2$ $T(e_2) = 2e_1+4e_2$ $T(e_2-e_1) = e_1 - e2$. What is the rank and nullity of T? Isn't this the rank and nullity of $e_1-e_2$? Is this asking what the rank and nullity of $[1 ...
1
vote
4answers
65 views

What to do with an empty column in the basis of the null space?

The problem says to find a basis of the null space of A, A being the matrix: $\begin{bmatrix}1 & 0 & 0\\1 & 0 & 1\end{bmatrix}$ So I need to solve the equation $Ax = 0$ to find the ...
0
votes
1answer
24 views

Is this an admissible singular value decomposition?

I would like to check if the following is an admissible SVD of the matrix: $$ \left(\begin{matrix} 3 & 1 & 1 \\ -1 & 3 & 1 \\ \end{matrix}\right) $$ ...
2
votes
1answer
158 views

Understanding the significance of row space and column space basis

I've just learned about the row and column space basis and I'm confused about what the significance of each is. My professor basically hasn't said much and has danced around any direct questions on ...
1
vote
2answers
98 views

Euclidean norm injective?

I seem to have thought myself into a corner. Can someone point out the hole in my reasoning here. Suppose $f:X \rightarrow \mathbb R$ where $f(x) = \|x\|,$ for $x$ in $X$. Knowing that $\|x\|=0$ ...
0
votes
0answers
40 views

Find a nonhyperbolic matrix which satisfies certain conditions

A nonhyperbolic matrix A for which the planar system $\overrightarrow x' = A \overrightarrow x$ has an equilibrium point at (0,0) with the x-axis as a stable curve and the y-axis as an unstable curve. ...
1
vote
2answers
46 views

Given a solution to find a matrix

For $e^{At} = 1/2\begin{bmatrix}e^{2t}+e^{-t} & e^{2t} - e^{-t} \\ e^{2t}-e^{-t} & e^{2t}+e^{-t}\end{bmatrix}$ for all t $\in$ $\mathbb{R}$. how to find A?
1
vote
1answer
161 views

Given that $J$ is a matrix in Jordan Normal Form, What is the Jordan normal form associated to $J^T$?

This seems pretty straight forward, but I'm not sure how to prove it. I have that If $A \in M_n$ and $B \in GL_n$, then $J = B A B^{-1}$, so $J^T = (BAB^{-1})^T$. So this give us that $J^T = B^T ...
0
votes
1answer
73 views

Direct Sum: Complement

Does every set admit a complement to form a direct sum as: $X=A\oplus A'$ I don't think so and my first guess would be: $X:= \mathbb R$, $A:= \lbrace -1,0,+1\rbrace$ Can somebody proof this? Or give ...
0
votes
1answer
59 views

finding the determinant of a matrix in terms of irreducible factors

I used a combination of row reduction and expanding along the rows to get the determinant as $\frac{1}{x}((x^2-yz)(x^2-zy) -(z^2-xy)(y^2-xz))$ however I can't seem to reduce it anymore, could ...
0
votes
0answers
36 views

How to define conditions under which linear maps are injective?

In this book (http://linear.axler.net/) proposition 3.2 states the following: Proposition 3.2: A linear map $T : V \rightarrow W$ from vector space $V$ to vector space $W$ is injective if and only if ...
5
votes
2answers
96 views

Is $\sqrt{1+x^2}$ matrix monotone?

A function $f(x)$ is matrix monotone if $f(A)-f(B)$ is positive semidefinite whenever $A-B$ is positive semidefinite for positive semidefinite matrices $A, B$. Is $\sqrt{1+x^2}$ matrix monotone?
2
votes
1answer
122 views

Proof involving matrix equation [duplicate]

$A$ and $B$ are $(n\times n)$ matrices and $AB + B + A = 0$. Prove that then $AB=BA$. How should I approach this problem?