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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Random multipliers of polynomial values at known points in $\mathbb{Z}_p$

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
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subspace of the Vector Space of real valued functions

This is a problem from Hoffman and Kunze's Linear Algebra 2nd edition. I am trying to determine whether or not a particular subset of the set of all real valued functions is a subspace. I've done ...
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34 views

find these linear functionals

I'm trying to solve this question: My attempt of solution: If $x\in E$, see $x$ in the first $m$ coordinates of $\mathbb R^n$ (can we do this?). I know how to find linear functionals such that ...
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Linear code from larger linear code

Question 2.16 of Essential Coding Theory by Guruswami, Rudra and Sudan asks to produce a $[n - d, k - 1, d'\geq\lceil d/q\rceil]_q$ code from an arbitrary $[n, k, d]_q$ code. Here we are working over ...
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Tensor product of algebras which is Frobenius.

Let $A$ and $B$ be two finite dimensional algebras over a field $k$. Let us suppose that the $k$-algebra $A\otimes_{k} B$ is Frobenius (or symmetric). Is it true that $A$ and $B$ are two Frobenius ...
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Prove that the determinant of a given matrix is proportional to the area of the triangle whose corners are the three points.

For three points in 2D, $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the determinant of \begin{bmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 2\\ x_3 & y_3 & 3\\ ...
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Proof check: any linear transformation can be represented as a matrix-vector product

I'm trying to prove that Theorem. Consider a linear transformation $T : \mathbb R^n \to \mathbb R^n$. The transformation $T$ can be represented as a matrix product $\mathbf x \mapsto A \mathbf ...
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100 views

Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
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157 views

Jacobian for a matrix transformation: Example of Cholesky decomposition

I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky ...
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84 views

Matrix partwise multiplication

I am working on an artificial intelligence application that (among other things) combines "opinions" of several "experts" who each have access to different aspects of a "situation". I can build this ...
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373 views

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
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111 views

Proof of rank-nullity via the first isomorphism theorem

I was thinking about the proof of the rank-nullity theorem and I thought about proving it as follows. I just wondered whether this proof worked? Lemma. If $V$ is a finite-dimensional $F$-vector space ...
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What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
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Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
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30 views

Finitely Generated Subset

Let $L\subset\mathbb{Z}^n$ be a subset closed under addition and subtraction. Can we show that $L$ is generated by at most $n$ elements?
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59 views

Show that a certain operator is symmetric

I am trying to prove that the operator $L^2 = -\partial_\theta^2 - \cot\theta\,\partial_\theta - \frac{1}{\sin^2\theta}\partial_\phi^2$ fulfills the following property: For $y_{l,m} = ...
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Integer Solutions To Linear Equation

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
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Find minimal set of cosets $C$, so that each $2$ vectors in $A_n$ are in one coset in $C$

Let $F_2^n$ be the set of all vectors of length $n$ with values of $0$ or $1$ and $A_n$ = $F_2^n \setminus(11\ldots1)$. Set $A_n$ contains all vectors except one with all $1$s. We can consider cosets ...
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206 views

How prove this stronger Cauchy-Schwarz inequality for traces of compression matrices

Question: Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
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Does anyone know any reference for this matrix?

For $n \geq 4$, $A$ is $(n-1) \times (n-1)$ tridiagonal block matrix $$A = n^2 \begin{bmatrix}B & -I & 0 & \cdots & \\-I & B & -I & 0 & \\ 0 & -I & B & -I ...
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Solve the matrix equation $[I+X(I-ST)^{-1}S](I-TS) = I$

Suppose that $I, S, T$ and $X$ are linear transformations of the vector space $W$, that $I$ is the identity and that $I-ST$ is invertible. Solve the equation $$[I+X(I-ST)^{-1}S](I-TS) = I$$ for $X$ ...
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maximizing the inverse degree in a graph

The inverse degree in the graph $G$ is defined as \begin{align*} r(G) = \sum_{i=1}^N \frac{1}{d_i}, \end{align*} where $d_i$ is the degree of node (vertex) $i$. Is the connected graph with maximum ...
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Extending generator matrix in coding theory

I have an $m \times n$ matrix $M$ where $m < n$ over finite field of size $2^w$. This matrix has the property that every $m \times m$ matrix formed by a $m$-subset of its columns is invertible. Is ...
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104 views

Cubic convergence of Rayleigh quotient iteration?

Trefethen and Bau, Numerical Linear Algebra, p. 208 states that Rayleigh quotient iteration (combining Rayleigh quotient estimate for eigenvalues and inverse power iteration) converges cubically ...
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Pythagorean like theorem for general spaces

There are laws, for example Pythagorean theorem, for calculating distance of two points, say $d(a,b)$, using third point and knowing $d(a,c)$ and $d(c,b)$ in vector spaces. My question is that for ...
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47 views

Linear Algebra progression in our times

I know that Linear Algebra is relatively new branch of Mathematics. I wonder, Is there any significant progress with Linear Algebra in our time? Are there any major questions which haven't solved ...
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76 views

Jordan normal form proof

In this proof of Jordan normal form in the Kaye and Wilson book, then for a transformation $T$ with minimal polynomial $m(x) = (x-e)^k$, they take a basis of $\texttt{ker}\;T$, extend it to a basis of ...
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105 views

How can we solve this question without brute force

If $A\in GL_n(R)$, where $R$ is a commutative ring with identity, I would like to prove $$ M=\begin{pmatrix} A & 0 \\ 0 & A^{-1} \\ \end{pmatrix}\in ...
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Problem about real square matrix with rank 1

Given $A \in \mathbb{R}^{n \times n}$ and $\text{rank}(A) = 1$. By working only on real field, show that $A$ is diagonalizable if and only if $\text{tr}(A) \neq 0$. Here, $\text{tr}(A)$ is the sum of ...
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77 views

Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { ...
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Eigenvectors of distinct eigenvalues of symplectic matrix on $K^{2n}$ orthogonal?

I showed that for the standard hermitian form $\langle , \rangle _{I_n}$ on $\mathbb{C}^n$ the eigenvectors of distinct eigenvalues of a matrix associated to this hermitian form are orthogonal to each ...
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The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
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50 views

How to prove this identity?

Let $$ x_1=- \frac{b\, t}{\left(a + e\, t\right)}, \\ x_2=- \frac{c\, s\, \left(a + e\, t\right)}{\left(a\, b + a\, f\, s - b\, h\, s\, t + e\, f\, s\, t\right)}, \\ x_3=- \frac{q\, \left(a\, b + a\, ...
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50 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
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Let $A$ be an $n$ x $n$ matrix with complex entries such that Trace$(A)$ = $0$. Then $A$ is similar to a matrix with $0$ in the diagonal entries

Let $A$ be an $n$ x $n$ matrix with complex entries such that Trace$(A)$ = $0$. Then how to show that $A$ is similar to a matrix with $0$ in the diagonal entries?
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(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
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50 views

Row operations that change similarity class

Let $\mathbb{K}$ be a field and $A\in \mathcal{M}_{n\times n}(\mathbb{K})$ be a matrix. Which row operations on $A$ do not change its similarity class?
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Is there a name for the group of complex matrices with unimodular determinant?

Does the group $$ G = \left\{ A \in \mathbb{C}^{n \times n} : |\det(A)| = 1 \right\} $$ have a name? It obviously contains the unitary group $U(n)$ and the special linear group $SL(n,\mathbb C)$. ...
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57 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
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Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
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An interesting system of equations

We have the following system with a and b, real numbers: $ax+y + z =4$ $x+2y+3z=6$ $3x-y-2z=b$ Show that $\forall a \in \mathbb{Z} $ there is a $b \in \mathbb{Z}$ such that the system admits a ...
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Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
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151 views

How can higher-dimensional projection maps be described mathematically?

New question: (resulting from discussions with Sabyasachi) I am wonder how can higher-dimensional projection maps, analogous to for example the Mercator, Miller, Behrmann projections, can be ...
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57 views

Condition for a particular matrix to be semi definite positive

Let $B$ a symmetric $N\times N$ real matrix whose diagonal elements are equal to one, that is to say $B_{i,i}=1$, $\forall i = 1, \dots N$ $B_{i,j} = B_{j,i}$, $\forall i,j = 1, \dots N$. $B_{i,j} ...
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Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
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300 views

What is the dual matrix (of a sample covariance matrix)?

Let $A$ be a matrix. I am most interested in the real, symmetric case, but for full understanding let's let $A$ be complex. What does it mean for $A^D$ to be the dual matrix of $A$? Can we interpret ...
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31 views

Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which ...
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58 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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49 views

What does the following matrix inequality mean?

Suppose $\{A_i\}_{i=1}^n$ is a collection of $m\times m$ matrices. I'm trying to understand the following criteria: There exists $\lambda\in[0,1)$ such that for all $x\in\mathbb{R}^m$ $$ \lambda ...
3
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115 views

Square root of positive definite nonsymmetric matrix

Let $N$ be a nilpotent matrix in $M_n({\mathbb R})$, such that $(I+N)^2$ is “positive definite” (but not necessarily symmetric) in the sense that $<X,(I+N)^2X>$ is positive for any nonzero ...