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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Why does this system of linear equations give me a linearly independent vector?

Given the vectors $\mathbf{v}_1 = \begin{pmatrix}0 \\ 1 \\ 2\end{pmatrix}, \mathbf{v}_2 = \begin{pmatrix}2 \\ - 1 \\ 4\end{pmatrix}$ we want to find a third vector $\mathbf{v}_3$ such that these three ...
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46 views

What is the intersection of row space and column space?

What is the intersection of row space and column space ? I calculated the row space and the column space of the matrix below (first the column space then I transposed it to find out the row space ...
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Examples of generalized geometric series.

I'm preparing a short presentation on the convergence of the geometric series of matrices, and I'd love some examples of their uses. I've encountered them when approximating inverses of matrices ...
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118 views

Proof that $\text{im}(g)^\top=\ker(f)^\top$ if $\text{im}(f)=\ker(g)$

Let $f:V\rightarrow W$ and $g:W\rightarrow X$ be two linear maps with $\text{im}(f)=\ker(g)$. How do I prove that $\text{im}(g)^\top=\ker(f)^\top$? I am allowed to use the fact that if $f$ is ...
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matrix version of Rayleigh quotient

Let $G$ be $m$ times $p$ matrix and $W$ be $m$ times $m$ symmetric positive definite matrix. Then, is the following true? $G'WG \le \lambda_{max}(W)G'G$ where $\lambda_{max}(W)$ denotes the maximum ...
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Finding basis B and C of a Linear Transformation WRT B and C

new to this site, so please forgive any formatting errors. :) I have come across a practice problem for a midterm, but there is no solution, so I'm not sure what I am supposed to do for the question: ...
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31 views

solutions for linear equation

Let there be: $$2\cdot a\cdot y+\left(a+4\right)\cdot x = 0$$ $$\left(2\cdot a+6\right)\cdot z+\left(16\cdot a-16\right)\cdot y+ \left(6\cdot a+24\right)\cdot x = 4\cdot a-2$$ ...
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47 views

Is Doobs theorem of binary rank really true?

The theorem states that any adjacent matrix of the line graph of a connected graph has a binary rank n-1 if the order, n, of the graph is odd. I have pondered about this and found that it doesn't ...
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27 views

Determinants of mixed matrices involving complex and non-commuting Grassmann variables.

I was working on a problem involving a mixed matrix of Grassmann variables and complex numbers. Essentially I found two expressions for a so called "superdeterminant" of the below matrix $G$. At the ...
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38 views

Finding the inverse of a matrix whose components are functions of t

I am to find the inverse of the following matrix: A(t) = \begin{bmatrix} t & t^4\\ 0 & 5t \\ \end{bmatrix} and after I used this formula to find the inverse The ...
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21 views

Calculate numeric solution to equations

If I know the following equations $$ x_1 + x_2 + \ldots + x_n = a_1 $$ $$ x_1^2 + x_2^2 + \ldots + x_n^2 = a_2 $$ $$ x_1^3 + x_2^3 + \ldots + x_n^3 = a_3 $$ $$ \ldots $$ $$ x_1^m + x_2^m + \ldots + ...
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Linearly extending a planar rotation to a spatial rotation

I'm currently reading the book "Finite Reflection Groups" by Grove and Benson for a kind of undergraduate course the Germans call "Proseminar" - it basically means I have to read up on a topic which ...
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31 views

Symmetric matrix in $\mathbb{Z}_{2}$× $\mathbb{Z}_{2}$

assume that $A_{n×n}$ is a symmetric matrix which all entries is in $\mathbb{Z}_{2}$ and all enteries of main diagonal is equal to 1. prove that there is a vector, name 'v' in $\mathbb{Z}_{2}^{n}$ ...
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why do odd magic squares have the same rank as their size?

why do odd magic squares have the same rank as their size whats special about odd magic squares? and why do even magic squares alternate The results are below where n is the size and r is the ...
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44 views

If $A \in \mathrm{GL}_n(R)$, $A^l=1$, what do we know about $\mathrm{char}(A)$, the characteristic polynomial of $A$?

Let $K$ be an algebraically closed field. Let $R$ be a (connected, if needed) $K$-algebra. Let $A \in \mathrm{GL}_n(R)$ such that $A^l=1$ for some $l$. Is it true that $\mathrm{char}(A)= ...
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35 views

Quadratic casimir of a representation of SO(N)

For $SO(N)$ the quadratic Casimir for the spinorial representation is $N(N-1)/8$ and that of the vector representation is $N-1$, but what is the quadratic Casimir of the spin $2$ representation? or ...
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63 views

Projection Theorem - understanding two parts of the proof

Projection Theorem. If $M$ is a closed subspace of the Hilbert space $H$ and $x\in H$, then (i) there is a unique element $x'\in M$ such that $$ \lVert x-x'\rVert=\inf_{y\in M}\lVert ...
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Determining linearized polynomial

I'm studying this article, and I have related to the proof of theorem 14. Let $f$ be a linearized polynomial over $\mathbb{F}_{q^m}$, which means $f$ is of the form $$ f(x)=\sum_{i=0}^{k-1} a_0 ...
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29 views

Null space for linear map injectivity?

The null space of a linear map $\mathcal A:U\to V$ is: $$ \text{Null}(\mathcal A)=\{u\in U|\mathcal A(u)=0_V\} $$ For $\mathcal A$ to be linear, we require that $\text{Null}(\mathcal A)=\{0_U\}$, ...
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42 views

Inverse of block triangular matrix

How to find the pseudo-inverse of the following block lower triangular matrix? $$X=\begin{bmatrix} A & 0 \\ c & d \\ \end{bmatrix}$$ Where $A$ is a $n\times n$ lower triangular matrix, $d$ is ...
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Find in terms of f and g, the projection of h upon S, where S = span{f,g}

My attempt so far is: ai) $\operatorname{Proj}_S(x)= Px$, where $P = (w_1w_1^T+w_2w_2^T)$ and $\{w_1,w_2,\ldots,w_m\}$ is an orthogonal basis for $S$. Let $w_1 = f$ and $w_2 = ...
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Finding all $x$ in $\mathbb{R}^4$

The answer I get for this matrix is: $$ x_3 \left[ \begin{array}{cc} 4\\ 3\\1\\0 \end{array} \right] +x_4\left[ \begin{array}{cc|c} 0\\ 0\\0\\1 \end{array} ...
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31 views

How to compute the change-of-coordinates matrix from B to B' with 2x2 matrices?

I have $$B=([1,1], [1,0])$$ $$B'=([0,1],[1,1])$$ and I need to find the change-of-coordinates from B to B'. I have performed the following steps: $$ \begin{bmatrix} 0 & 1 | \ 1 & 1\\ 1 ...
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Clebsch-Gordan Coefficients for 8 in $3 \otimes \bar 3 $ and the $6$ in $3 \otimes 3$ of $SU(3)$?

Do Tables for the Clebsch Gordan coefficients for the decomposition of the $8$ dimensional irrep of $SU(3)$ into $3 \otimes \bar 3 $ and the $6$ in $3 \otimes 3$ (in the Dynkin basis) exist somewhere? ...
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24 views

Simple vector/matrix norm questions

(1) If I have a column vector $x$ and it's transpose $x^T$ (a row vector), can I say $$\|x^T\|_2\|x\|_2 = \|x\|^2_2$$ (2) Additionally, am I correct in believing that $\|x^T\|_2\|B\|_2 \geq ...
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69 views

How to categorize Measures, Topologies, Algebraic Structures, etc

I'm trying to make a big picture of mathematics as a concept so that I get a better understanding of it and can connect different parts of it and understand the relationship between them. In the ...
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41 views

fast computation of traiangular matrix with certain pattern

I was wondering for a matrix with the following form. \begin{bmatrix}p_0&0&0&0\\p_1&p_0&0&0\\p_2&p_1&p_0&0\\p_3&p_2&p_1&p_0\end{bmatrix} Is there any ...
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Without determinants, show a set in $L^n \subset K^n$ is linearly independent over $K$ if it is over $L$.

Let $L$ and $K$ be fields with $L \subset K$. Let $v_1,\ldots,v_r \in L^n$ be column vectors, linearly independent over $L$. Of course, we can also consider the vectors to sit in $K^n \supset L^n$. ...
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Why do we divide out by commutators for the trace operation?

Given a unital ring $R$, it's easy to form the ring $M_n(R)$ of square $n\times n$ matrices over $R$. Any such matrix has a trace, which is simply the sum along the diagonal. However, I've come to ...
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126 views

Characteristic polynomial of adjoint

I'm trying to show that the adjoint transformation $T^*$ of the endomorphism $T$ on a finite dimensional, real inner product space has the same characteristic polynomial as $T$ in a coordinate free ...
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Express $S$ in set notation and determine whether it's a subspace of the given vector space $V$.

$V = R^{n}$, and $S$ is the set of all solutions to the non-homogeneous linear system $Ax=b$, where $A$ is a fixed $m × n$ matrix and $b(≠0)$ is a fixed vector. Express $S$ in set notation and ...
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22 views

Exponential of the product of a scalar with a complex symmetric $2 \times 2$ matrix

Say I want to compute $e^{t M}$, for $M=\left[\begin{array}{cc} a & b\\ b & a \end{array}\right]$ with $a, b \in \mathbb{C}$ and $t \in \mathbb{R}$. Can I absorb $t$ into $M$ and do as ...
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63 views

Geometric Descriptions of Kernel and Range of T

I have a pretty straightforward question. First, I was given a linear transformation of T : R^2 to R^2. I found that the dimension of the kernel was 1 and the dimension of the range was 1. This is ...
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33 views

Are there always integer solutions to this special case of $Ax \ge b$?

The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$. The output is a matrix $A(n\times n)$ ...
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Prove that a vector $v \in \ker T$ if and only if $[v]_{B_V} \in \operatorname{null} A$

Let $V$ and $W$ be vector spaces over a field $F$ with $\dim V = n,$ $\dim W = m$ with bases $B_V = (v_1,\ldots,v_n)$ and $B_W = (w_1,\ldots,w_m),$ respectively. Assume $T:V \to W$ is a linear ...
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In proving trace(AB) = trace (BA), I not only swapped summands but also the order of summation - is this ok to do?

In proving this for a Linear Algebra course, I'm guessing we are allowed to assume that the vector spaces are finite-dimensional, so a double summation, summing up to n, does not pose a problem, when ...
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Is a symmetric positive definite matrix similar to a triadiagonal symmetric positive definite matrix?

Is any real symmetric positive definite matrix similar to a triadiagonal symmetric positive definite matrix? From Householder lemma it is known that if $B$ is symmetric, there exists a orthogonal ...
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Relation between eigenvector and image of linear transformation.

Let $\phi$ be the linear transformation from vector space V to itself. (dimV = n) By choosing certain basis set, I can write it as a matrix form, say M Then I want to know that If {$v_1$,...$v_p$} ...
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52 views

Rank nullity failure

I'm doing a problem with the matrix $$A=\left[\begin{array}{cccc} 0&3&3&3\\0&0&0&0\\ 0&1&0&1 \end{array}\right].$$ I found that the column space has dimension two ...
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29 views

Linear system of equations including block-matrices

Consider $Q$ which is a $NK \times NK$ symmetric positive definite ( or semidefinite) matrix, partitioned symmetrically into blocks $Q_{ij}$ which are $K \times K$ where we know that all $Q_{ii}$'s ...
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Computing columns of a pseudo-inverse

I need to compute the pseudo-inverse of a very large rectangular dense matrix without any special structure or properties. I run out of memory/computing power and have no access to a large parallel ...
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29 views

Rotation matrices and angular velocity

Let us assume I have an object O with axis $x_{O}$, $y_{O}$, $z_{O}$, with different orientation from the global frame S with $x_{S}$, $y_{S}$, $z_{S}$ (I don't care about the position). Now I know ...
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32 views

Computing the trace and determinant of a matrix representation relative to a real basis, from the matrix relative to a complex basis,

Let $A$ be the complex matrix representing a transformation of the vector space $C^2$ of 2-tuples over the complex numbers into itself, relative to the natural ordered basis {(1,0),(0,1)}. Let $A_R$ ...
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How do we solve for $\frac {\partial x} {\partial \|x\| }$ ?

Here $x \in \mathbb{R}^N$ and $\|.\|$ is the standard $l2$ norm. Is this the element-wise inverse of $\frac {\partial \|x\| } {\partial x}$ ? I tried to work this out element wise but nothing came of ...
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31 views

Maximizing over a p-Frobenius norm ball

Let $p\in [1,\infty]$, $p^*$ be s.t. $1/p+1/p^* =1$ (when $p=1$, $p^*$ is understood to be $\infty$, and vice versa). $||u||_p :=(\sum_{i=1}^n |u_i|^p)^{1/p}$ is the p-norm for vectors in ...
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34 views

Isomorphisms of normed vector spaces

I'm having difficulty with the following question: Show that, for $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ and for an infinite discrete time-domain $\mathbb{T}$, $\exists$ an isomorphism of normed ...
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45 views

Dimension of homogeneous polynomials passing through 4 points

Can anyone help me solve the following exercise? Let $p_1,p_2,p_3,p_4$ be distinct points in the projective plane $P^2$. What is the dimension of the vector space of homogeneous polynomials ...
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Find a polarisation in a complex vector space.

I have trouble with the following problem Let $V$ be a complex vector space, of real dimension $2d$, and let $L$ be a lattice in $V$, that is the abelian group generated by a basis of $V$ as a real ...
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34 views

How does permuting rows of a positive matrix affect the eigenvectors?

Assume I have a positive matrix (strictly positive entries) $A$. By perron-frobenius theorem, spectral radius (positive by definition) is indeed the largest eigenvalue (called perron root) with the ...
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21 views

Adding an additional vector to a vectors space $U$, prove $U=span(v,u_{1},\ldots u_{n})$

I'm posed with the following statement to prove: Let $V$ be a vector space over $\mathbb{F}$ and define $U=span(u_{1},\ldots,u_{n})$, where for each $i=i,\ldots,n$, where $u_{i} \in V$. Now ...