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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246 views

Trace of a matrix times outer product

$\DeclareMathOperator{\trace}{tr}$Is there any relationship between $\trace(Sxx^T)$ and $x^TSx$? Is there a nice way to write the set of quadratic functions of the components of a vector $x$ given ...
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2k views

Rotating one 3-vector to another

I have written an algorithm for solving the following problem: Given two 3-vectors, say: $a,b$, find rotation of $a$ so that its orientation matches $b$. However, I am not sure if the following ...
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1answer
207 views

Columns of matrix $A_{m \times n}$ span $\mathbb{R}^{m}$

I am having trouble understanding the following theorem given in my textbook: Let $A$ be an $ m \times n $ coefficient matrix. Then the following statements are logically equivalent (That is, for ...
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5answers
128 views

Looking for proof of $\sum_1^n \alpha_i x^i = 0 \;\; \Longrightarrow \;\; \alpha_i = 0, \forall\;i$.

The assertion in the subject line is an abbreviated form of: $$\sum_{i=1}^n \alpha_i x^i = 0, \forall \; x \in {\mathbb R} \;\; \Longrightarrow \;\; \alpha_i = 0, \forall\;i \in \{1, \dots , n\}.$$ ...
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3answers
71 views

Linear algebra ( orthonormal basis)

$U$ is subspace for $\mathbb{R}^3$ with orthonormal basis $u_1,u_2$. Given $v\in \mathbb{R}^3,\;$ let $a_1=\langle v,u_1\rangle ,\;\; a_2=\langle v,u_2\rangle$ So it must be the case that: If ...
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4answers
166 views

linear algebra question

Consider the system \begin{align*} x + y + 2z &= 2 \\ 2x + 3y - z &= 5 \\ 3x + 4y + z &= b \end{align*} (a) For what values of $b$ does the system have a solution? Using this value of ...
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2answers
165 views

Finding the trace of a block matrix

Let $A$ be a $5 \times 5$ skew-symmetric matrix with entries in $\mathbb{R}$ and $B$ be the $5 \times 5$ symmetric matrix whose $(i, j)^{th}$ entry is the binomial coefficient $\binom{i}{j}$ for $1 ...
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0answers
20 views

conjugate space

Find the coordinates $\phi^\ast(\epsilon_{2}^\ast+\epsilon_{3}^\ast)$ in base $A^\ast$ where $A$ is $(3,4,5),(1,3,-2),(0,1,-2)$ and $M(\phi^\ast)_{st}^{st}=\begin{bmatrix} 1 & 3 & 2 ...
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2answers
102 views

Let $V$ be the inner product space consisting of linear polynomials

I am stuck with the following problem: Let $V$ be the inner product space consisting of linear polynomial, $p\colon[0,1]\rightarrow\mathbb R$ (i.e. $V$ consists of polynomials $p$ of the form ...
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2answers
192 views

Standard Basis of the Finite Field of Prime Numbers

A little info regarding this field: Addition and multiplication in $Z^n_p$ behave as usual but with the remainder taken upon division by $p$. Ex: $Z_3$ will only consist of the three integers ...
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1answer
47 views

Least-squares solution to an over-defined probem.

This problem arose in my stereo vision project. I have two matrix equations: $$\left( \begin{array}{ccc} x_1.w_1 \\ y_1.w_1\\ w_1 \end{array} \right) = \left( \begin{array}{ccc} A_{11} & ...
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1answer
71 views

The addition of linearly independent vectors.

If we say that $u$ and $v$ are linearly independent vectors in the vector space $V$ over some field $\Bbb F$, then $u+v$ is a linearly independent vector. Why is this? Isn't it possible for $u+v=0_v$ ...
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1answer
15 views

How to rearrange $(pgh_b)(s^2)-(pgh_t)(s^2)$ to $(pg)(s^2)(h_b-h_t)$

can someone please show me the steps to go from this $$ (pgh_b)(s^2)-(pgh_t)(s^2) $$ to this $$ (pg)(s^2)(h_b-h_t) $$ Thank you.
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2answers
251 views

Exponent of $GL(n,q)$.

Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group. ...
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1answer
1k views

Converting parametric equation to implicit form

So I have the equation defined in homogeneous coordinates $[w; x, y]$ as $[1+t^2; 1-t^2, 2t]$ $$w = 1+t^2$$ $$x = 1-t^2$$ $$y = 2t$$ If I do $w+x-y$ I get $-2t+2$, so $t = -(w+x-y-2)/2$. I was then ...
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70 views

Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?

The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element is not adjacent to the diagonal, then there are ...
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2answers
242 views

How to workout the determinant of the matrix $D_n(\alpha, \beta, \gamma)$.

I am going through an example in my lecture notes. This is it: Let's introduce the matrix $D_n(\alpha, \beta, \gamma)$, which looks like this: $$\pmatrix{\beta & \gamma & 0 & 0 ...
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1answer
1k views

Need help with a linear equation with a free variable?

Is $a_1, a_2, a_3$ a linear combination of $b$? $a_1 = (1, -2, 0), a_2 = (0, 1, 2), a_3 = (5, -6, 8), b = (2, -1, 6)$ I used Gaussian elimination to get to. $$ \left[ \begin{array}{@{}ccc|c@{}} ...
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2answers
100 views

Functional in the dual space s.t $\psi$ restricted to a subspace equals $\phi$

Let $W$ be a subspace in a vector space $V$ [you may assume that dim $V$ is finite]. Prove that any $\phi$ in $W^*$ can be extended to a functional on $V$ , i.e. there exists $\psi$ in $V^*$ such that ...
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2answers
950 views

$V$ is isomorphic to $V^{\ast\ast}$, the double dual space of $V$.

Prove that for any vector space $V$ the map sending $v$ in $V$ to (evaluation at $v$) $E_v$ in $V^{**}$ such that $E_v(\phi) = \phi(v)$ for $\phi$ in $V^*$ , is injective. Derive from this that if ...
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249 views

Inverse Function Theorem/ Polynomial

I was thinking about this after I read about Jacobian conjecture. But I can't see what I did wrong? Maybe you can help me. Let $F: \mathbb{C}^n \to \mathbb{C}^n$ be of the form $F(x_1, \dots, x_n)= ...
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2answers
119 views

A matrix algebra question

I have the following situation, $$E = A X B + C X D$$ where $A,B,C,D,E$ and $X$ are matrices with proper dimensions. I want to obtain an expression like, $$X = f(A,B,C,D,E)$$ i.e., leave $X$ alone at ...
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124 views

If $Ax=b$, for $b\ne 0$, has more than one solution, then $Ax=0$ does as well. T or F. Prove this.

I get that this is true, because there's one free variable, so no matter what the augmented matrix is, there always will be an infinite amount of solutions. Right? But how to I explain this as a ...
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3answers
51 views

determinant $s=n+1$

Need help $A=\begin{vmatrix} s&s&s &\cdots & s&s\\ s&1&s &\cdots & s&s\\ s&s&2 &\cdots & s&s\\\vdots & ...
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1answer
80 views

dimension of subspace hermitian matrix

Let $T\in H(m+1;\mathbb{R})$ be a symmetric matrix, s.t. $T^2=T$ and $tr(T)=1$. Now let $U:=\{ S\in H(m+1;\mathbb{R}): TS+ST=S\}$ be a subset of the vector space of all hermitian matrices. I want to ...
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112 views

Construct an orthonormal basis

Consider vector space of functions continuous on $I \subseteq \mathbb R$ and scalar product is defined as $({\bf f}, {\bf g}) \equiv \int_{I} \rho(x) f(x)g(x) dx$. Let the generating function $G(z,x)$ ...
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0answers
34 views

solving for two points, having 4 defined, using linear graph as connector

I have been on it for few days now, maybe someone could tell me I am on the right track. So i got 2 points of a linear graph: Ax, Ay Bx, By And a ...
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3answers
155 views

Intuition about Hyperplane

I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?
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2answers
275 views

Can Every Square Matrix be written as product of two commuting matrices.

The title explains it all. Can every square matrix $A$ be written as $A=B_1B_2=B_2B_1$ of any two matrices $B_1$,$B_2$.
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70 views

“unitary group” with respect to non-hermitian matrices?

the group $GU(n,q)$ is usually defined as the group of $n$ by $n$ matrices $X$ over $\mathbb{F}_{q^2}$ such that $X^H X=I_n$, where $I_n$ denotes the identity matrix and $^H$ the hermitian transpose ...
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86 views

Prove that $\mathbf{A}^2 - \mathbf{A} + \mathbf{E} = \mathbf{0}$ $\implies$ matrix $\mathbf{A}$ is regular

Show that if a matrix $\mathbf{A}$ satisfy $\mathbf{A}^2 - \mathbf{A} + \mathbf{E} = \mathbf{0}$ then matrix $\mathbf{A}$ is regular. (Note that $\mathbf{E}$ denotes identity matrix.)
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1answer
52 views

find z such that $\vec{u}$ and $\vec{v}$ are linearly independent

The problem is given as, "Determine the values of z such that the vectors $\vec{u} = \pmatrix{-1\\z}$ and $\vec{u} = \pmatrix{z\\-1 + z}$ are linearly independent. Here is my work... $\pmatrix{-1 ...
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1answer
265 views

householder transformation matrix

Hi could you help me with the following: Let A be the matrix $$\pmatrix{-2 & 1& 1 \\ -2& 2& 1\\2 &-2& 3 \\ }$$ with an eigenvalue $\lambda = 2$ and corresponding ...
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1answer
99 views

Proof involving homogeneous system of linear equations with det 0.

This is from Hoffman & Kunze: Consider $Ax = 0$ with $A = \bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$. Such that $ad - bc = 0$ With some element of A nonzero. Then ...
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2answers
252 views

Dimension of sum of 3 subspaces

Let $W_1$, $W_2$ and $W_3$ be finite-dimensional subspaces of a vector space. Show that it may happen that $W_i \cap W_j = 0$ for all $i \ne j$, but still $\dim(W_1 + W_2 + W_3) \ne \dim W_1 + ...
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1answer
243 views

List of various vector (linear) spaces [closed]

The vector (linear) space is defined as a non-empty set L over a field F, where two relations (binary operations) are defined: Addition $ \oplus: L \times L \longrightarrow L $ Scalar ...
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148 views

are there any “deep” reasons for representing linear systems as $Ax=b$ instead of $xA=b$?

Nowadays we represent the system of $m$ linear equations $$\sum_{i=1}^na_{1i}x_i=y_1$$ $$\sum_{i=1}^na_{2i}x_i=y_2$$ $$\vdots$$ $$\sum_{i=1}^na_{mi}x_i=y_m$$ as $\mathbf{Ax}=\mathbf{y}$, where ...
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1answer
117 views

Prove $\det(A+I)=1$

Need help with my homework. $A \in M_{nxn}(\mathbb{R})$ is upper-triangular and $A^{n}=0$ Please hint how to prove, that $\det(A+I)=1$ I dont know how it do, know laplace equation
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431 views

basis of a matrix that is transposed.

So lets say the transposed matrix is this: Span{(0,-3,6,6,4,-5),(2,1,4,2,8,-9),(3,7,-5,-8,8,9),(3,9,-9,-12,6,15)} The RREF for the original matrix is $$ \begin{pmatrix} 1 & 0 & 3 & 2 ...
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1answer
44 views

linear transformation dimZ

What i should do? I have to find $\dim (Z)$. I have a map $\psi(( x_{1},x_{2},x_{3},x_{4}))=(x_{1}+2x_{2}-x_{3}+3x_{4} , 2x_{1}+5x_{2}+x_{3}+7x_{4}, x_{1}+x_{2}-4x_{3}+2x_{4}) $ And $\displaystyle ...
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3answers
73 views

Square of two positive definite matrices are equal then they are equal

I have read that if $P, Q$ are two positive definite matrices such that $P^2=Q^2$, then $P=Q$. I don't know why. Some one can help me? Thanks for any indication.
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1answer
24 views

showing the solution linear map is particular solution + solution of homogeneous sytem

Let $AX=B$ be a system of linear equations, where $A$ is $m\times n$ matrix and $X$ is $n$-vector, and $B$ is $m$-vector. Assume that there is one solution $X=X_0$. Show that every solution is of ...
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1answer
43 views

base depending on the parameter

Need help $V = \text{lin}((1,1,3,2),(4,5,2,5),(2,3,-4,1),(1,2,-5,5))$ a)Find base and $ dimV$ b)For which $ t \in \mathbb{R}$, exist base $\alpha1, \alpha2, \alpha3, \alpha4 \in \mathbb{R}^4$ such ...
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48 views

Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq

For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm? -I can easily show the ...
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143 views

Help understanding train problem

A train $150$ $m$ long passes a km stone in $15$ seconds and another train of the same length traveling in opposite direction in $8$ seconds. The speed of the ...
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1answer
48 views

how to show that the set of mutually perpendular unit vectors is basis of vector space

Let $V$ be a finite dimensional vector space over $\Bbb R$, with positive definite scalar product. Let $\{v_1, v_2, ... v_m\}$ be the set of elements of $v$ of norm $1$. and mutually perpendicular. ...
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1answer
78 views

Find column number from cell position in a table

Known variables: Number of rows, number of columns, number of cells, cell index. Find: column number relative to cell index. Example: 1 2 3 4 5 6 7 8 9 What ...
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1answer
315 views

Computation of the minimal polynomial of a matrix in Mathematica

On the wolfram website, the following program is given for computing the minimal polynomial of a square matrix $a$ in the variable $x$: MatrixMinimalPolynomial[a_List?MatrixQ,x_]:=Module[ { ...
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0answers
746 views

what is dimension of orthogonal complement of a subspace of a vector space.

This is last part of my other question. I don't understand the last part of problem. Feel free to edit the question. c) Let $V$ be a vector space of real $n \times n $ symmetric matrices, what is ...
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0answers
150 views

Is there a simple interpretation of the eigenvectors of a graph (visualizable?)?

I want to understand eigenvectors obtain from graphs (adjacency matrices) in an analogous way as they are interpreted from principal component analysis of a set of images, which is easy:Eigenfaces ...