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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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 ...
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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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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\\ ...
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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 ...
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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 ...
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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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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 ...
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105 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\\ ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 = ...
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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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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 $\pi=V \rightarrow V_i$. ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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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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 $\mathbb{C}$ ...
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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 ...
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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. ...
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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 ...
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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 ...
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28 views

Are these theorems the same?

I'm studying adjoint operators from Schaum's book and I'm confused with these theorems: So the author proves the conjugate transpose $B^*$ is the adjoint of $B$. But some lines after, he states in ...
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What is the the projection of vector b onto the matrix A if b is in the Column space of A?

What is the the projection of vector b onto the matrix A if b is in the Column space of A? This is a strange question for me. Can you do a projection in this situation?
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Splitting an Indefinite Matrix into 2 definite matrices

I'm attempting to use some quadratic programming techniques to solve a particular optimization problem and my chosen Objective Function is indefinite. I've found some texts online which regard ...
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Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
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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} = ...
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Proving that $\mathrm{rank}(P_1+P_2) = \mathrm{rank}(P_1)+\mathrm{rank}(P_2)$

Supposing $P_1$ and $P_2$ two projectors as: $P_1\circ P_2 = P_2\circ P_1$. What is the condition for $P_1+P_2$ to be a projection? If it was the case above then how can I prove that ...
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1answer
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Linearly Independency of vectors [on hold]

Are the vectors $(e^{\frac{\pi}{2}},1)$ and $(110^{\frac{1}{3}}, 1)$ in $\mathbb{R}^2$ linearly independent?
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How do I prove the adjoint matrix is the adjoint of the operator

We know that $A^t$ is the adjoint of $A$ if we are in the euclidean spaces: $\langle Au,v\rangle=(Au)^tv=u^tA^tv=\langle u,A^tv\rangle$ (where $\langle u,v\rangle=u^tv$) I couldn't prove the ...
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Prove that $sgn(\sigma_1 \circ \sigma_2) = sgn(\sigma_1)sgn(\sigma_2)$

Lete $n\in \mathbb{N}$. Show that the transformation $$sgn: S_n \rightarrow \{\pm 1\}$$ (where $S_n$ is the set of all permutations of the integers in the set $\{1,...,n\}$),given by $\sigma \mapsto ...
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Show which matrices are upper triangular orthogonal in $\mathbb R$.

Show which matrices are upper triangular orthogonal in $\mathbb R$. I've tried written the matrix product $Q^T Q$ and I get the following equations: $x_{1,1}^2 = 1$ $x_{2,1}^2 + x_{2,2}^2 = 1$ ...
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Show that the image is spanned by the columns of the matrix

No idea how to attempt this, was in an old exam paper for my linear algebra class
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35 views

These inner products don't match in $\mathbb C^n$

In $\mathbb C^n$, we can define the inner product between $u=\{u_1,\ldots,u_n\}$ and $v=\{v_1,\ldots,v_n\}$ as $\langle u,v\rangle=u_1\overline{v_1}+\ldots+u_n\overline v_n$. I've read in a book that ...
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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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Solution of tridiagonal system

I need a method for solving a system $(x_1 \ldots x_n)$ of type A*x=B where A is a tridiagonal matrix of type \begin{equation}\begin{bmatrix} x & x & \cdots & \cdots\\ x & x & x ...
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T-cyclic subspaces and dimensionality?

For each linear operator T on vector space V, find an ordered basis for the T-cyclic subspace generated by vector z. V=R^4 T(a,b,c,d)=(a+b,b-c,a+c,a+d) z=e1 (e1 is the first standard basis in R4 ...
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Find two bilinear forms with the same quadratic form over $\mathbb F_2$

Let $V$ be a $K$-vectorspace with a bilinear form $\langle , \rangle$ and the associated quadratic form $q:V \to K, v \mapsto \langle v,v \rangle$. Let $K = \mathbb F_2$. Are there two different ...
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Eigenvalues of the vectors of

I came across the following problem: "Let $\mathbf a\in\mathbf{R}^n$ be a fixed $n$-component real, non-zero, vector. Let $A^+$ and $A^−$ be real $n\times n$ matrices with components: $$(A^\pm)_{ij} ...
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47 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
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19 views

Formula needed for calculating probability of recurring events

I'd like to find an answer for calculating the following recurring events: You have X opportunities of picking a ball from a sack. Every time after a ball is picked, the ball is returned to the sack. ...