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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26 views

Can I modify a polynomial to return only multiples of a given number?

I'm attempting to create a polynomial equation for a project of mine, with a shape similar to the following: $${3x^5\over500}+{x^4\over25}+x^3+40 x^2+100 x$$ However, one of my goals is to have the ...
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0answers
10 views

Is this a sufficent answer to - Similar matrices have same Rank

A justification along the lines of, Matrices that are similar are representations of linear maps w.r.t different basis and therefore have the same rank Is that an acceptable proof? or would I need ...
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0answers
16 views

Confused about the association between two vectors

Polynomial $x^3 + 2x^2 + 4 \in P_3(\mathbb R)$ and $(1, 2, 0, 4) \in \mathbb R^4$. $x^3 + 2x^2 + 4$ is equivalent to $(1, 2, 0, 4)$ apparently because $(1, 2, 0, 4)$ $= 1 (1, 0, 0, 0) + 2 (0, 1, ...
5
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4answers
2k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
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1answer
495 views

Least square solution based on the pseudoinverse solved efficiently with singular value decomposition

Hi apologies it's hard to type out the problem, I have a lecture slide on neural networks. It says the fitting error gives the matrix: N by M matrix of thi's multiplied by Mx1 weights minus Nx1 ...
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0answers
11 views

Inner Product inequality problem using Cauchy Schwarz, or what other way?

Let $<p,q>$ be an inner product on n. If p and q are both of degree n, show that $<p,q>^2$ $\leq$ $<p,p>$ $<q,q>$. I tried multiplying the right side out but am getting ...
1
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1answer
15 views

Multicollinearity and SVD

I compute the Singular Value Decomposition of a n x n matrix. If the matrix is not full rank, and I have 2 collinear columns, I end up with one singular value equal to 0. Is it possible to find out ...
3
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1answer
30 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
5
votes
1answer
389 views

Anti-commutative matrices

If $A$ and $B$ are anti-commutative square matrices, so $AB+BA=0$, how do you a) prove that $\mathrm{tr}(A)=\mathrm{tr}(B)=0$ and b) prove that the order of the matrices is even?
4
votes
1answer
278 views

Spectral radius and positive definite of matrices

Denote $ \rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
2
votes
2answers
58 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
5
votes
4answers
208 views

if $AB\neq 0$ for any non zero matrix $B$ then $A$ is invertible

Question is to check that : If $A$ is an $n\times n$ matrix over a field $F$ and $AB\neq 0$ for any non zero matrix $B_{n\times n}$ over $F$ then, $A$ is invertible. This does make some sense to me ...
1
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1answer
31 views

Proof of Eckart-Young-Mirsky theorem

Could someone please explain why in http://en.wikipedia.org/wiki/Low-rank_approximation#Proof_of_Eckart.E2.80.93Young.E2.80.93Mirsky_theorem it says "we know that $\exists(k+1)$ dimension space ...
0
votes
1answer
14 views

Compute the vector $v$ if the coordinate vector $[v]_{s}$ is given with respect to each ordered basis $S$ for $V$

Ok, so this is a practice question in my book: $V$ is $M_{22}$ $S=$ \begin{bmatrix} 1&-2\\ 0&0\\ \end{bmatrix} \begin{bmatrix} -1&3\\ 0&1\\ \end{bmatrix} \begin{bmatrix} 1&0\\ ...
3
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0answers
28 views

If $A \in M_{n,n}(\mathbb F)$ is invertible then $A = UPB$, $U$ is unipotent upper triangular, $B$ is upper triangular and $P$ is a permutation.

If $A \in M_{n,n}(\mathbb F)$ is invertible then $A = UPB$, where $U$ is unipotent upper triangular, $B$ is upper triangular and $P$ a permutation matrix. A hint is given that one could relate ...
3
votes
1answer
210 views

Column and Row Picture for Singular System of 100 Equations (Strang P55, 2.2.32)

Start with 100 equations $\color{#8F00FF}{A}\mathbf{x} = \mathbf{0}$ for $\mathbf{x} = (x_1, ..., x_{1oo})$. Suppose elimination reduces the 100th equation to $0 = 0$, so the system is "singular". ...
1
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1answer
19 views

A is a Hermitian projection if and only if it is an orthogonal projection

I need to figure out this property of Hermitian / Orthogonal projections "A is a Hermitian projection if and only if it is an orthogonal projection" Your assistance will be highly appreciated. ...
2
votes
1answer
78 views

Valid Proof for Cayley Hamilton Theorem? (Not the usual incorrect one)

By induction; case n=1 is true. $A$ admits an eigenvalue $\lambda$ with eigenvector $v$ over $\mathbb{C}$. Change $A$ into a basis $e_1=v,...,e_n$. Then $\exists X$ such that $XAX^{-1}=\left( ...
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3answers
58 views

An infinite generating set of a finite dimensional vector space contains a basis

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ? Please help
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4answers
28 views

The row rank of an $m\times n$ matrix $A$ is at most $\min\{m,n\}$. Why?

Ok, so let $A$ be an $m\times n$ matrix. I understand by intuition that the row rank has to be $\le m$, but why also $n$? Is this because there can be no more leading ones than $m$ or $n$?
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0answers
18 views

Least squares polynomial approximation $(f-p_n,q)=0$

I know how to do the other way around but I am getting stuck with showing the following If $(f-p_n,q)=0$ then $p_n$ is a polynomial of best least squares approximation in a norm $|\cdot|$ for a ...
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0answers
11 views

Odd and Even Weight functions in orthogonal polynomials proof

Suppose now that w is an even function, i.e. $w(-x)$ = $w(x)$ for all x in $[-1,1]$ and let $p_0$,..., $p_n$ be a family of orthogonal polynomials with respect to w. Prove by induction that $p_k$ is ...
4
votes
1answer
37 views

Canonically isomorphic but not equal

In mathematics, we have many objects that are canonically isomorphic but not equal on the nose. For example let $V$ and $W$ be vector spaces. Then $V\otimes W$ and $W\otimes V$ are canonically ...
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4answers
111 views

How to prove a set of vectors does not span a space.

Ok, so I'm a bit curios as to how you can prove a set does not span a vector space. For example, let ${S}$ be the vector set \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0\\ ...
6
votes
2answers
93 views

Why this formula is positive definite?

I have a formula $A(I+GQ)^{-1}(G+GQG)(I+QG)^{-1}A^{\mathrm T}+G$ where $A,Q,G,I\in\mathbb R^{n\times n}$, $A$ nonsingular, $G$ positive semi-definite, $Q$ positive definite, $I$ the identity matrix, ...
1
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0answers
20 views

Show $W_1 \hookrightarrow V \twoheadrightarrow W_2$ is an isomorphism

Let $\langle , \rangle$ be a non-degenerate bilinear form with the signature $(p,q)$ on a real vectorspace $V$ and $W_1, W_2$ subspaces, such that the restriction $\langle , \rangle |_{W_i}$ is ...
0
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0answers
23 views

Find Determinant of linear transformation

The question is Find the determinant of linear transformation Let V be the vector space of polynomials of degree at most over R, and define T:V to V by T(p(x))=p(1+x)-p'(1-x) for all p(x) in V. I ...
0
votes
2answers
20 views

proving a natural projection is linear and finding its kernel

Let $V_i = 1,...,N$ be a collection of vector spaces over a field $F$. Consider the Cartesian product $V=V_1 \times V_2 \times ... \times V_N$ with the natural projections $\pi=V \rightarrow V_i$. ...
1
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1answer
22 views

Let $A$ be a symmetric $n\times n$ matrix and suppose that $A$ is positive definite. Then $a_{jk}\leq$ $\frac 12(a_{jj}+a_{kk})$.

Let $A$ be a symmetric $n \times n$ matrix and suppose that A is positive definite. Then $a_{jk}$ $\leq$ $\frac 12(a_{jj}+a_{kk})$. Can somebody please explain whether it is True or False? Thanks ...
0
votes
1answer
28 views

Integral solutions to $Ax = y$

What is a necessary and sufficient condition that the solutions of $Ax = y$ be integers whenever the components of $y$ are integers, given that the elements of $A$ are integers? When $A$ is ...
0
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0answers
22 views

Proof about dual bases?

Let V be a finite dimensional vector space over a field F. Let B={v1,v2, ..., vn} be a basis and consider the dual basis B*={v1*,v2*,...,vn*}. Let a be an element of V*. prove that $$v = ...
0
votes
2answers
15 views

Show that a linear operator P is orthogonal [on hold]

inner product (A|B) = tr(A B^t) linear operator A(X) = X^t Is P skew or self adjoint? self = P = P* neg is skew
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1answer
23 views

prove following properties of self-adjoint operator

$A: V \rightarrow V$ self-adjoint; $b$ is a real number. Show 1) the minimal polynomial has distinct roots; 2) $\ker(L) = \ker(L^k)$ for $k\geq1$; 3) $\text{im}(L) = \text{im}(L^k)$ for k bigger ...
0
votes
1answer
55 views

Why is the Det(a)=0 not a subspace? [on hold]

I'm reading my linear algebra textbook, and it says word for word: The following sets is not a subspace when the set of all 2x2 matrices B such that det(B)=0. I just need help trying to understand ...
1
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3answers
62 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
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2answers
17 views

Finding an Orthonormal Basis using Gram Schmidt

Given the set of vectors $S=${${V_1=\binom{1}{4},V_2=\binom{4}{-4} }$} I am to find an orthonormal basis for $R^2$ using the Gram-Schmidt process. I've already worked it out and found the orthonormal ...
2
votes
1answer
18 views

Inverse of Cartan matrix

The Cartan matrix of the root system $A_n$ looks like, denote it by $A'_n$ $$A'_n= \begin{bmatrix} 2 & -1 & 0 & 0&\ldots & 0 \\[0.3em] -1 & 2 & -1 ...
0
votes
0answers
20 views

Check if a vector b is orthogonal to column space of A

Using built-in matlab functions, how would you check if a vector b is orthogonal to the column space of matrix A given that the dimensions of A and b are correct and given that b is not in the column ...
2
votes
1answer
26 views

Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x} $ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x} $ For what values of $a$, $E1$ and $E2$ have ...
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0answers
10 views

generalisation of Knonecker matrix product

In the Kronecker matrix product $C = A\otimes B$ we have that $C(i,j)=A(i,j)*B$ where the elements $A(i,j)$ are just numeric scalar values. What if the $A(i,j)$ are matrix operators which act on ...
0
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1answer
25 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
0
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1answer
57 views

Which of the following functions $f : \mathbb{R^2} → \mathbb{R^2}$ is a linear transformation?

Which of the following functions $f : \mathbb{R^2} → \mathbb{R^2}$ is a linear transformation? So I've cross out b and d since they do not work with the zero vector. But both a and c look like they ...
0
votes
0answers
15 views

Matrix Transformations On a Point to Create Fractals

I am working with $3X3$ matrices to perform operations on 2 dimensional geometries, in this a case a 2-D point represented by a $3X1$ matrix. Where the third coordinate is homogeneous. I wish to ...
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2answers
33 views

How would I express the statement “Let H be a subspace of V” in mathematical notation?

How would I express the statement "Let H be a subspace of V" in mathematical notation? Does something like this work? $$ ( \ \ H(\mathbb{R})\subset V(\mathbb{R}) \ ) $$
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2answers
57 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
1
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0answers
17 views

Under what conditions on the field k will all symmetric matrices be diagonalizable?

It's a theorem that if $A$ is an $n \times n$ symmetric matrix ($A = A^{T}$) with real entries, then $A$ is diagonalizable. The proof goes like this: $A$ has a complex eigenvalue, since $\mathbb{C}$ ...
0
votes
2answers
28 views

Help with this easy lemma of linear algebra

I'm trying to demonstrate a theorem of linear algebra and I need to prove this lemma to finish the proof: Let $A=(a_{ij})$ be the matrix representation $T:V\to V$ in the orthonormal basis ...
0
votes
0answers
24 views

Determinant, Rank

Lat $K$ be a field, $K\subset \Bbb C$. $a_0,a_1,a_2,\dotsc$ is a sequence, $a_i\in K, i=0,1,2,\dotsc$ For integers $s,m\geq0$, Defined $$A_{s,m}=\begin{bmatrix} a_s & a_{s+1} & \dotsc ...
1
vote
1answer
36 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
8
votes
0answers
54 views
+200

Can a cube always be fitted into the projection of a cube?

If we project the unit cube in $\mathbb{R}^n$ onto a $k$-dimensional subspace of $\mathbb{R}^n$ which contains the origin, can we always fit a $k$-dimensional cube of side length 1 into the ...