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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find a matrix transform

Given a vector $v={(v_1,v_2,...,v_n)}^T$, I would like to find some matrix operations on $v$ to create an $n \times n$ matrix $X$ such that its entry $X_{i,j} $ satisfy (1), (2), (3), (4), ...
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123 views

Odd town Even town explanation.

I am struggling to understand the solution to the following problem: If $\mathcal F\subset 2^{[n]}$ such that for each $F_1$ and $F_2$ in $\mathcal F$ we have $|F_1|,|F_2|\equiv 1 \bmod 2$ and ...
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56 views

Solving Multiple Equations with Many Variables

Here's a problem I have stumbled upon, which may have a straightforward solution with linear algebra. If so, I cannot see it. Choose $n > 0 \in \mathbb N$, and consider the sequence of equations: ...
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100 views

Scalar-by-matrix Derivative of Quadratic Product

I'd like to know $\frac{\partial f(\mathbf{U})}{\partial \mathbf{U}}$, i.e., the 'by-matrix derivative' of the following scalar function $f(\mathbf{U})$ w.r.t. $\mathbf{U}$. $$f(\mathbf{U}) = ...
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53 views

Question about Universal Mapping Property

Now I seem to understand the construction of the tensor product of $R$-Modules by defining $$F_R(M \times N) := \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$ and constructing the submodule $D$ in ...
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29 views

Rank problem in inversion of t(A) %*% A in R

I need to get the inverse of the cross-product $(\mathbf{A}' \mathbf{A})$, and I run into numerical problems that don't make any sense to me. I actually need $(\mathbf{A}' \mathbf{W}^{-1} ...
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47 views

How to show that $ \phi(x)(xI-A)^{-1}= M_0x^{n-1}+ M_1x^{n-2} + \cdots + M_{n-2}x + M_{n-1}$

How do I show that $$ \phi(x)(xI-A)^{-1}= M_0x^{n-1}+ M_1x^{n-2} + \cdots + M_{n-2}x + M_{n-1}$$ where $A$ is a $n$ dimension square matrix, I is an identity matrix, $\phi(x)=x^n + a_1 x^{n-1} ...
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55 views

Infinite series for recurrence

Question 1 If I define $A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} \tag 1$ (where $a_n$ are $3\times 3$ constant matrices indexed with n), then can we re-write $\sum_{n \ge 1} a_{n-1} \frac{z^n}{n!} ...
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101 views

$m \times n$ matrix where $m < n$

So I'm a long distance student and I need some help to bounce ideas off of other people who understand the work. Fellow students are few and far between. So while this is an assignment question, I ...
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39 views

Base of Subspace with vectors

Let E be the vector subspace of $R^3$ generated by it vectors $v1 = (1,2,0)$ and $v2 = (-1,0,2)$ How can find a basis of E between the following vectors? $$w1=(-2,-12,8), w2=(-12,-2,-8), ...
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101 views

If $A \succeq B$ is it true that $B^{-1} \succeq A^{-1}$

If $A$ and $B$ are two positive definite matrices such that $A - B$ is nonnegative definite, is it true that $B^{-1} - A^{-1}$ is positive definite? The doubt came to me when working with confidence ...
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40 views

The largest real eigenvalue of a matrix is bigger than 1

I have a problem which is interesting: given a real matrix $A_{n\times n}$, when this matrix has a largest real eigenvalue which is strictly bigger than 1. If possible, can you give some conditions ...
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62 views

question about inner product and $f^*$

In $\mathbb{R}$3 we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$ we have operator $f ...
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66 views

Inner product of functions as integration

I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties. My ...
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165 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
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39 views

Prove another matrix is positive definite given that A is a Hermitian matrix

Suppose that $A$ is a Hermitian symmetric $n\times n$ matrix of complex numbers all of whose eigenvalues lie inside the interval $(-1,1).$ Prove that the matrix $A^3+Id$ is positive definite. An ...
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34 views

Diagonal matrices and integrals

Suppose that $$A=\int_{\alpha}^{\beta} f(B,x)\ dx,$$ where $B$ is a $3\times3$ matrix. The result I'm looking for is that if $B$ is diagonalized with an orthogonal matrix, then is A diagonalized by ...
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111 views

Simple proof that a $3\times 3$-matrix with entries $s$ or $s+1$ cannot have determinant $\pm 1$, if $s>1$.

Let $s>1$ and $A$ be a $3\times 3$ matrix with entries $s$ or $s+1$. Then $\det(A)\ne \pm 1$. The determinant has the form $as+b$ with integers $a$,$b$ and it has to be proven that $a>0$ if ...
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102 views

Sequences length for LFSR in the general case

An LFSR with a reducible polynomial can generate several sequences, depending on the initial value. My goal is to have an algorithm to compute those length without going through the enumeration of all ...
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35 views

Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
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56 views

Counterexample of $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$

I know that $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$, where $T^{*}$ means the adjoint operator of a linear operator $T$, holds when the domain of $T$ is finite-dimensional. However, the proof uses ...
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186 views

Strange proof of Schwarz Inequality with Pythagorean Theorem

Does anyone know what is going on in this proof of the Schwarz inequality? Most importantly: how can one assume that $c^2\leqq \|A\|^2$, or later on, that $c^2\|B\| \leqq \|A\|^2$? This would imply ...
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26 views

Determining a spanning set for $X/\bigcap_{i=1}^N \ker{\lambda_i}$, where each $\lambda_i$ is a linear functional on $X$

Let $X$ be a vector space over a field $K$. Suppose that $\{\lambda_i\}_{i=1}^N$ is a collection of linear functionals $\lambda_i : X \to K$. Let $W$ be the subspace $\{ x \in X \mid \lambda_i(x) = ...
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31 views

permutations with a given condition!

What will be the number of permutations of n different things, taken r at a time,when p particular things is to be always included in each arrangement? I know the answer to this question but could not ...
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37 views

Find the Jacobian of F

Given that $A \in \mathbb{R}^{m\times n}$, and $b \in \mathbb{R}^{m}$, we define: $$F:\mathbb{R}^{n} \rightarrow \mathbb{R} = \left\| Ax-b \right\|^2$$ Find the Jacobian of $F$, and show that it is of ...
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59 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
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66 views

Orthogonal Operator Infinite Dimensional Inner Product Space

I know that on a finite dimensional inner product space, a unitary (or orthogonal) operator preserves the inner product. That is, having $\|T(x)\|=\|x\|$ for all $x\in V$ is equivalent to having ...
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177 views

Sequences length for LFSR when polynomial is reducible

An LFSR with polynomial 1+x4+x5 = (1+x+x2)(1+x+x3) can generate several sequences, depending on the initial value. If I did not made any mistake enumerating them, the sequences length are 3, 7 and 21. ...
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40 views

Fixed Matrices over finite field by a map

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ ...
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76 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
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74 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
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36 views

Prove Basis for symmetric matrix.

**Let V be the vector subspace of M$_{2}$ ($\mathbb{R})$ consisting of all symmetric matrices, That is A$^{t}$ = A. 1) Show that $\clubsuit$= $\left\{ \left(\begin{array}{cc} 1 & -2\\ -2 & 1 ...
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69 views

Rotations - linear or quadratic?

In linear algebra rotations are represented by matrices, i.e. linear transformations How do you formally prove that rotation is a linear transformation? But this page is very interesting ...
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56 views

Dimension of a vector space.

Let $v_1,v_2,v_3,v_4$ and $v_5$ be the non-zero vectors of a vector space $V$ such that $a_1v_1+a_2v_2+a_3v_3+a_4v_4+a_5v_5\neq0 \hspace{1cm} (\forall a_i\neq0,\, 1\leq i\leq5)$ Then what is the ...
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96 views

R-linear functionals on manifolds

Surely the following is well known: Let $X$ be a (differentiable) manifold, $R$ the ring of continuous/smooth real functions on $X$, $V$ the $R$-module of all continuous/smooth vector fields on ...
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128 views

How do you find a non zero vector in Linear Algebra?

The question is; The vectors $a_1 = (1, 1, 0)$ and $a_2 = (1, 1, 1)$ span a plane in $\Bbb R^3$. Find the projection matrix P onto the plane, and find a nonzero vector $b$ that is projected to zero. ...
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49 views

Direct Sum of Three Subspaces

Suppose $U = \{(x, y, x+y, x -y, 2x) \in \Bbb F^5 : x, y \in \Bbb F\}$. Find three subspaces $W_1, W_2, W_3$ of $\Bbb F^5$, none of which equal $\{0\}$ such that $\Bbb F^5 = U \oplus W_1 \oplus W_2 ...
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37 views

Decomposing a square matrix into two non-square matrices

I have a matrix $A$ with dimensions $(mxm)$ and it is positive definite. I want to find the matrix $B$ with dimensions $(nxm), (n << m)$, which follows the following expression: $$A = B'B$$ Here ...
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92 views

Linear Algebra - Question about transformation and characteristic polynomial

I have some trouble with this question, I tried to solve it but I'm not sure that my solution is correct. I'll be glad if somebody could take a look. Data : T : R^4 --> R^4 (linear transformation) ...
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44 views

Calculating amount of rotation to straighten an imaginary line created by 2 points.

I am trying to build a small app where my users can straighten up a tilted face with just 2 clicks I ask my users to click on the middle of the nose and the middle of the eyebrows of the face ...
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48 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
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114 views

Proof Verification, Uniqueness of vector $v$ satisfying $\varphi(u) = \langle u,v\rangle $ for a linear functional $\varphi$.

I want to prove the uniqueness of the following vector $v$. The existence of the vector is guaranteed. We know that there exists at least one vector $v$ for every $u$ such that for a linear ...
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34 views

Transposing Map on SO(n) matricies

Does there exist a matrix X such that we can describe the transposing of an SO(n) matrix R as something like $M \rightarrow XMX^T = M^T$ I am particularly interested in SO(n) in even dimensions, ...
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112 views

Find max and min subject to constraint ||x|| = 4

$Q(x,y)=7x^{2}+12xy+12y^{2}$ I only know how to do this is $\|(x,y)\|=1$ If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$. I don't know what to do if ...
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40 views

$(Ax,x)>0 \forall x$ implies $A$ selfadjoint?

I read in Reed & Simons's Functional Analysis (Vol.1, pg. 194) that a positive operator, i.e, an operator $A$ such that $(Ax,x)>0 \forall x$ on a complex Hilbert space is selfadjoint, but this ...
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131 views

linear transformation of finite dimensional vector spaces

Let $V$ and $W$ be finite dimensional real vector spaces and $T\colon V\to W$ be linear. (a)Prove that if $\dim(V) < \dim(W)$, then $T$ cannot be onto. (b)Prove that if $\dim(V) > \dim(W)$, ...
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72 views

Gram-Schmidt: Do the sets have some sort of order?

I'm learning about the Gram-Schmidt process: I have some subspace basis $A$ with three vectors: $$A = \{a_1,a_2,a_3\}.$$ Based on it, we will create an orthonormal basis $B$ with three vectors, ...
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30 views

Equivalence of sets

Let $u_1, u_2, u_3 \in \mathbb{C}$ be the cubic roots of unity I'm wondering if the following two sets (balls) are equivalent: $$ \lbrace (v,w) \in \Bbb C^2 : \vert v \vert + \vert w \vert \leq 1 ...
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53 views

why can identity matrix sometimes be trivially determined by context?

For any matrix $A$, $A I = I A = A$ ($I$ is the identity matrix). If $A$ has $m\times n$ dimension, the first identity matrix $I$ that appears in the above equations should have $n\times n$ ...
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69 views

Show that $\langle x, Ax \rangle + \langle b, x \rangle = c$ can be transformed to $\langle x', Ax' \rangle = 1$

Let $A$ be a real, regular, symmetric $n \times n$ matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$ How can I show that $$\langle x, Ax \rangle + \langle b, x \rangle = c$$ can be transformed by ...