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 ...

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Schur's Triangularization Lemma in Hefferon's Linear Algebra textbook

I'm reviewing some material and came to this: Fix a basis $B = \{\vec{\beta}_1, \ldots, \vec{\beta}_n\}$ for $V$ ($V$ is a vector space) and observe that the spans $$ [\emptyset] = \{\vec{0}\} ...
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If $A \in M_{3\times 3}(\mathbb{R})$ is normal, there is an orthogonal $O$ such that $O^TAO$ is either diagonal or in this form

Given an $n \times n$ normal matrix $A$ over $\mathbb{R}$, show that there is an orthogonal matrix $O$ such that $O^TAO$ is either diagonal or in the form ...
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If $W,W^\perp$ are invariant under $A\in M_{n\times n}(\mathbb{R})$, $char(A)=\mu_A=\mu_{A|W}\mu_{A^T|W^\perp}$

The full exercise is as follows: Let $W$ be an invariant subspace of a matrix $A \in M_{n\times n}(\mathbb{R})$. Let $\mu_A$ be the characteristic polynomial of $A$. Prove the following: (a) ...
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What is the orthogonal complement

Let $\langle A,B \rangle = \text{tr}(B^TA)$. For $M_n(\mathbb{R}$): Find the orthogonal complement of the diagonal-square-matrices. So I need to find $$U^{\perp} = \left\{ A\in M_n(\mathbb{R}) \mid ...
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a question about normal matrices

Let $A$ be a normal matrix and $\lambda$ a scalar. Show that $A-\lambda I$ is also a normal matrix. $A$ is a normal matrix then there is a unitary diagonalization of $A$ over $\mathbb{C}$. ...
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Finding basis for image and kernel of A = [x+y,x-y,x+z,x-z,x+y+z]

Here is a problem I'm trying to solve: Knowing that vectors $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \Bbb{R}^{2013}$ are linearly independent, find basis for image and kernel of ...
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Is it possible to simplify this expression?

Recently I've been stuck in an expression: $$\sum_{m=1}^{N}{{\lambda}_m {\textbf x_m}^T}\cdot \sum_{m=1}^{N}{{\lambda}_m {\textbf x_m}}$$ where $\textbf x_m\in {\mathbb R}^n$ is a column vector. Could ...
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A=B.C if C is non-singular then col(A)=col(B) [closed]

Let A,B,C be matrices such that A=B.C if C is non-singular, then col(A)=col(B). is this the case?
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73 views

Existence of adjoint of the inverse

Let $H$ be a Hilbert space over $\mathbb{F}$ and $V$ be an inner product space over $\mathbb{F}$. Let $T:H\rightarrow V$ be a bounded linear bijection. If $V$ is a Hilbert space, then the open ...
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What does it mean to find a matrix of linear transformation in given basis?

I have a following problem to do: A linear transformation $f: \Bbb{R}^3 \rightarrow \Bbb{R}^2$ is defined with a formula: $$f(\mathbf{x}) =( \begin{smallmatrix} x_1+x_2\\ x_2+x_3 ...
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what is an example of a normed space such that $||\sum x_i||=\sum ||x_i||$ does not imply they have the same direction?

Let $V$ be a normed space over $\mathbb{K}$ and $x_1,...,x_n\in V\setminus\{0\}$ such that $||\sum_{i=1}^n x_i||=\sum_{i=1}^n ||x_i||$. If $||\cdot||$ satisfies parallel law, then this implies that ...
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76 views

Matrix reduction trigonalisaton

Let $ \mathbf{A}=\begin{bmatrix} 2 & -1 & -1 \\ 2 & 1 & -2\\ 3 & -1 & -2 \end{bmatrix} $ Trigonalise a matrix in process of ...
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Given polynomials $f,g$ on an algebra $F$ and operators $T,D$ corresponding to integration/differentiation, show that $T[(Tf)g] = (Tf)(Tg) - T[f(Tg)]$

My question is as to whether there is an elegant way of demonstrating the desired result, without going through the nontrivial amount of computations required to attain the desired result. The ...
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Finding the basis and dimension of a vector space

Find the basis and dimension of vector space $ L_{1}$ spanned by vectors $ a_{1} ,a_{2},a_{3} $, the basis and dimension of vector space $ L_{2}$ spanned by vectors $ b_{1} ,b_{2},b_{3} $ and also ...
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138 views

How close apart are two message - “Document Distance” algorithm

I was looking at this algorithm that computes how close apart are two texts from one another and the formula seems a bit weird to me. The basic steps are: For each word encountered in a text you ...
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Self-adjoint matrices: prove that $\operatorname{Tr}\left((AB)^2\right)\le\operatorname{Tr}\left(A^2B^2\right)$ [closed]

$A,B \in M_n(\mathbb C)$ and self-adjoint. Prove the following inequality: $\operatorname{Tr}\left((AB)^2\right)\le\operatorname{Tr}\left(A^2B^2\right)$. Thanks
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$P$ is an matrix invertible Proof $|\lambda I-PBP^{-1}|=|\lambda I -B|$

I have this problem : $P$ is an matrix invertible Proof : $|\lambda I-PBP^{-1}|=|\lambda I -B|$ I'm not so sure about my answer, since I don't think I could use "double" determinant for example ...
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Maple: How do I type “solve” with an arrow under?

I am trying to learn using Maple 18 (Mac). I have defined a function with a list of X and Y values. f := x->LinReg(X, Y, x) Now I would like to output the unknown "x" value that correlates with ...
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Finding inverse linear transformation

I'm solving a homework question and I'm stuck with it's last part. The question goes like this: Let $\displaystyle T:M_n(\mathbb{R})\to M_n(\mathbb{R})$ be a transformation defined as ...
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93 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. ...
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106 views

Tensor Product: Vector Spaces

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

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 ...
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How can a $k\times (k-m)$ matrix be multiplied by a $k\times m$ matrix?

While reading a book on differential geometry, I came across this line: Since the differential $d\psi_0(x_0):\mathbb R^m\to \mathbb R^k$ is injective, there is a matrix $B\in \mathbb R^{k\times ...
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Why then is $df(x)$ surjective if and only if $Ax\neq 0$?

$\forall x\in \mathbb R^k$, define the linear map $df(x):\mathbb R^k\to\mathbb R$ as follows:$\forall \xi\in \mathbb R^k, df(x)(\xi)=2x^TA\xi$, where $A$ is symmetric. Why then is $df(x)$ surjective ...
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$\left\Vert J(x)^{-1}\right\Vert<2\left\Vert J(x^*)^{-1}\right\Vert. $?

Could you please help me to prove this theorem: Suppose $J:{\bf {\rm R}}^m\rightarrow{\bf {\rm R}}^{n\times n}$ is a continuous matrix-valued function. If J(x*) is nonsingular, then there exists ...
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63 views

Determinant of nXn matrix

I know this was already asked before here: Q: The determinant of a NxN matrix? But I still did not manage to solve this with the method he suggested. I tried adding all the columns to the first one ...
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64 views

Linear Algebra: Polynomials Basis

Consider the polynomials $$p_1(x) = 1 - x^2,\;p_2(x) = x(1-x),\;p_3(x) = x(1+x)$$ Show that $\{p_1(x),\,p_2(x),\,p_3(x)\}$ is a basis for $\Bbb P^2$. My question is how do you even go about proving ...
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Solving linear differential equations system

Upon trying to solve this particular system , I've encountered a few problems. $$ y'=5y+4z $$ $$z'=-4y-3z$$ After solving for eigenvalues the quadratic yielded a double root at $\lambda=1$ . But I ...
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Question in regards to definition: finite dimensional

Do we denote a vector space as finite dimensional IF it has a basis, or do we say that it is finite dimensional if it's associated through an isomorphic transformation with a "number space", ie. ...
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Why is the projection of a vector V onto a span W, independent of the orthogonal basis of W.

Very straightforward question. I have read time and again in my book that it is independent but I don't understand why? Wouldn't changing the basis mean changing the length of the projection?
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Linear maps that are matrices

If I have the linear map $A:\Bbb{R}^3\rightarrow \Bbb{R}^3$ where $A$ is a matrix. Is the matrix $A$ (along with the vectors it operates on) in a basis or not? I think it is not, since the vectors it ...
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checking if some vectors span $R^3$ that actualy span $R^3$

If we want to check if the following set of vector span $R^3$ (1,0,0) (0,1,-1) (0,4,-3) (0,2,0) then we forme an augmented matrix formed by the vectors which form the columns of the augmented matrix ...
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Solving variables in a matrix for a specific determinant

The matrix is as follows: $$ A = \begin{pmatrix} 0 & x & 1 & 2 \\ x & 1 & 1 & x \\ 1 & x & x & 1 \\ 1 & x & 1 & x \end{pmatrix} $$ What I want to do ...
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An exponential map of a matrix computation

Suppose the $n\times n$ matrices $A$ and $M$ satisfy $AM+MA^{T}=0.$ Show by direct computation that the product $\mathrm{exp}(At)~M~\mathrm{exp}(A^{T}t)=M$ for all $t\in \mathbb{R}.$ Note: By ...
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How many square root matrices?

I would like to see a proof of the following statement: A positive-semidefinite matrix has precisely one positive-semidefinite square root, which can be called its principal square root. I ...
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61 views

Can a unitary matrix be reducible?

Given an $n\times n$ unitary matrix $U$, is it possible for $U$ to be reducible? That is, can $U$ be transformed via a permutation transform $P$ such that the transformed matrix $PUP^{-1}$ is in upper ...
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Find an orthogonal basis for W.

Use the standard Euclidean inner product on $\mathrm R^4$. Let $W$ be the subspace of $\mathrm R^4$ spanned by $u_1 = (1, 1, 1, 1),$ $u_2 = (2, 4, 1, 5),$ $u_3 = (1, -5, 4, -8).$ Find an ...
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Finding basis for orthogonal subspace

Find a basis for $S^\perp$ for the subspace $$ S = span\left\{\left[\begin{matrix}1\\1\\-2\end{matrix}\right]\right\} $$ How do I start this question?
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The main involution on $ M_{2}(F) $ and it's extension to $ M_{2}(F_{\mathbb{A}}) $.

I'm presently reading through a paper of Shimura's; "Special Values of the Zeta Functions Associated with Hilbert Modular Forms". In the paper he defines $ \iota $ to be the main involution of $ ...
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Help with a homework problem involving $\textbf{H}$-conjugate vectors

My problem is the following: Let $\textbf{H}$ be a symmetric $n\times n$ matrix. Are the following claims true? Why? a) If the vectors $\textbf{d}_1$ and $\textbf{d}_2$ and vectors ...
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Let $\varphi:G\rightarrow H$ be a homomorphism, show $\varphi':G\rightarrow\text{im}(\varphi)$ is an epimorphism

Let $\varphi:G\rightarrow H$ be a homomorphism, show $\varphi':G\rightarrow\text{im}(\varphi)$ is an epimorphism Epimorphism is a surjective homorphism. We know that $\text{im}(\varphi)\subseteq H$ ...
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can Legendre polynomials take on different forms?

My prof. assigned some homework that has us compute legendre polynomials, but I'm getting polynomials that are different from ones that I reference with on like Wolfram and Wikipedia. I think the ...
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Show that $\{(1-a,1+a), (1+a,1-a)\}$ is linearly independent

In order to show that $$\{(1-a,1+a), (1+a,1-a)\}$$ is L.I. I did: $\beta_1(1-a,1+a)+\beta_2(1+a,1-a) = (0,0)\implies\\\begin{cases}\beta_1-\beta_1a + \beta_2 + \beta_2a = ...
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finding the jordan canonical with one eigenvalue

Let $$A= \left[ \begin{array}{ c c } -2 & -3 & 1 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{array} \right] $$ Determine the Jordan canonical form of A. The only ...
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If matrix $\sum_0^\infty C^k$ is convergent, how can I prove that $A(\sum_0^\infty C^k)B$ is convergent?

For an $n \times n$ matrix $C$ and If $\sum_0^\infty C^k$ is convergent, how can I prove that for two matrices $A$ and $B$, $A(\sum_0^\infty C^k)B$ is convergent? It seems quite obvious that you just ...
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Compute $\mathbb{Z}/m\otimes\mathbb{Z}/n$ using exact sequence

I want to compute $\mathbb{Z}/m\otimes\mathbb{Z}/n$ using exact sequence as follows. Consider the exact sequence $$ \mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/m\to 0. $$ Tensoring with $\mathbb{Z}/n$ gives ...
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How does the dimensions of the Kronecker product work out here?

According to http://en.wikipedia.org/wiki/Kronecker_product If we have linear operators $A,B$ representing linear maps $ S : V → X$ and $T : W → Y,$ then the kronecker product $A \otimes B $ has the ...
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Find orthogonal matrices

Let $A=\begin{bmatrix} 1 & -1/2&-1/2 \\ -1/2 & 1& -1/2\\ -1/2&-1/2 &1 \end{bmatrix}$. Is it possible to find explicitly orthogonal matrices $P, Q$ such that ...
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representation of a matrix norm of the inverse of a matrix

Let $A$ be an $n$ by $n$ nonsingular matrix and suppose that a matrix norm $|||\cdot|||$ is induced by the vector norm $\lVert \cdot \rVert$ on $\mathbb{C}^n$. Show that $$|||A^{-1}||| = ...
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Prove that $S(W)$ is Invariant subspace

Let $S, T: V\to V$ such that $ST=TS$. Let $W\subseteq V$. Prove that if $W$ is invariant subspace of $T$ then also $S(W)$ is invariant subspace of $T$. Let $w\in W$. $$T(S(w)) = S(T(w)) = S(w')$$ ...