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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Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
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Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
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Nonnegative solution to underdetermined linear system

I would like to show that the underdetermined system $Ax=b,\; x\ge 0$, with $b$ being a positive vector and $A$ being a binary matrix, has at least one solution. I've seen several other related ...
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1answer
35 views

How do I show that $\inf\limits_{\det(X)\neq0}\|X^{-1}AX\|^{2}_{F}=\sum\limits_{\lambda\in{\Lambda}}|\lambda|^{2}$?

Show that $$\inf\limits_{\det(X)\neq 0}\|X^{-1}AX\|^2_F=\sum_{\lambda\in\Lambda}|\lambda|^{2}$$ holds, where $\Lambda(A)$ is the set containing all eigenvalues of A, and $\|\cdot\|_{F}$ is the ...
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1answer
93 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ ...
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Jordan normal form book

I am currently reading the book Basic Algebra [modern] Anthony W. Knapp about Jordan canonical form Is there any detailed oriented book about Jordan Normal Form which explain : An Algorithm to put ...
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1answer
15 views

Solving a system of xor equations?

How can I solve the following system of xor equations? k0 ⊕ k2 ⊕ k3 = 0011 k0 ⊕ k2 ⊕ k4 = 1010 k0 ⊕ k1 ⊕ k2 ⊕ k3 = 0110 How can I solve this system to know the ...
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1answer
45 views

what does $p(-1) = 0$ mean?

In a linear algebra problem, it asks me to determine the subespace spanned by $$ \left\{ p(x) \in \mathbb{R}^3 : p(-1) = 0 \right\}. $$ What does it mean?
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In $\mathbb{R}^{3}$, does an orthogonal basis of integer vectors exist such that none of their coordinates is $0$?

In $\mathbb{R}^{3}$, does an orthogonal basis $\{$ $(a_{1}, a_{2}, a_{3}),$ $(b_{1}, b_{2}, b_{3}),$ $(c_{1}, c_{2}, c_{3})$ $\}$ exist such that all $a_{i}, b_{i}, c_{i}$ are integers $\neq 0$?
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Find the distance of a point from a plane generated by two given vectors

I need to calculate the distance of the point $P = (0, 5, -4)$ from the plane which pass from the point $P1=0, 1, -2)$ and generated by the two vectors: $$ v1 = (1, 2, 3), v2 = (-1, \sqrt{2}, 1) $$ ...
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30 views

Finding linear transformations. [on hold]

Can somebody provide some idea on how to tackle this problem? Find all linear transformations $$T:\mathbb{R}^3\to\mathbb{R}^2,\ (u,v,w)\to(x,y)$$ which map the $u-w$ plane onto the line given by the ...
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22 views

Characteristic Polynomial Calculation

I have a problem in my homework in which I have to find the characteristic polynomial of the following matrix: I know the final solution is: However, my answer keeps getting wrong whenever I ...
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1answer
303 views

Determine homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $y = 2x – 6$

Determine the homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $ y = 2x – 6$. I do $mx - y +b =0$: $\text{slope} = m$, $\tan(O)= m$ ...
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2answers
90 views

Probability of building an Invertible Matrix

If we build a 10X10 matrix,randomly filling with 1's and 0's, is it more likely to be invertible or singular? First of all until we have 10 1's its not going to be invertible. With 10 1's on the ...
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I need help in Algebra. [on hold]

can someone explain to me the Vector space and the Vector base? (their uses and how to find a matrix' base and space and so on..) thanks in advance
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Characteristic polynomial of a linear endomorphism of dimension $n$.

So, if $T: V \rightarrow V$ and I suppose that $T^2-3T+2I=0$ and that the $\mathrm{rank}(I-T)=k$, what would be the characteristic polynomial of $T$? I know from previous questions that the ...
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24 views

Jordan normal form Upper or lower

I am reading a jordan form book at the moment, Basic Algebra [modern] Anthony W. Knapp page $231$, but I feel the lack of understanding : should we have to start with the Bigger Jordan blocks of ...
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1answer
19 views

Vector Cross Product and Expression for perpendicular distance between any two Vectors

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||} $$ where $\times$ means the vector cross product. Attempt: ...
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3answers
40 views

Property of SO(3)

Suppose $A\in SO(3).$ Show that there exists a vector $v\in \mathbb{R}^3$ such that $Av=v$. $ SO(3)={{A\in O(3)|detA=1}} $ and $ O(3)={A:\mathbb{R}^3\rightarrow ...
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1answer
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Vectors components that are not contra or covariant?

I know that a vector can have contravariant and covariant components, but is it possible to have components that are neither contravarient or covariant? I suspect that the answer is yes, and that most ...
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$Ax = b$ & $Ax + b$

Ask a dumb question but confuse me long time. The following is what I know: 1st case $Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$. 2nd case $ f(x) = Ax + b$ ...
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If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$ .

Having trouble solving this. If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$. .
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How to find the long range transition matrix L of P

P is the transition matrix of a regular Markov chain. Find the long range transition matrix L of P. $$ P = \begin{bmatrix} 1/2 & 1/4 & 1/4\\1/2&1/2 &1/4\\0 &1/4 & ...
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Blocks in a layer? $X$ layers of blocks in a triangle, which $Y$ being the total number of blocks… (w/o using triangular numbers)

I have $32$ layers of blocks in a triangle. However, for the sake of variation, let's call that $X$. I have $1000$ square blocks total. We'll call those $Y$. The first layer of the triangle has $1$ ...
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1answer
34 views

matrix vs vector span {} linear algebra

I am in a University Linear Algebra course and am confused by the term span and its relation to both matrices and vectors. Can someone help clarify what they mean? =Span= Can it only be made of ...
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1answer
38 views

Linear subspace of K[X]?

I got the following task from my professor and I wanted to ask for advice from you. Task: $K$ is a field I shall prove this statement Prove that for every $v$ element of $K$ the set $I_v = ...
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1answer
17 views

Prove that the columns of the first matrix span but the columns of the second matrix do not span.

A = [1 0 1 0] = row 1 [1 2 0 1] = row 2 (2 * 4 matrix) and [0 0] = row 1 [2 1] = row 2 (2 * 2 matrix) I know that Column of matrix of m*n dimension spans if rank of matrix is equal to m. ...
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What are the standard defintions of “counterclockwise” and “clockwise” in 3d space?

I'm in Calc III right now, and I'm a little confused as to what constitutes "clockwise", and "counterclockwise" rotations when dealing with the various planes in 3d-space. Of course, it's obvious in ...
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396 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
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1answer
25 views

Clarifications of problem with parameters: the relationship between matrices and endomorphism

Let $f$ be an endomorphism of $R^3$ such that $f(a,b,c)=(2b,a-b,b)$. I don't understand how I can see for which values of $k\in R$ there esist $$\begin{pmatrix}-2 & 0 & 0 \\ 0 & k & 0 ...
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2answers
76 views

$x=x^2$ in a sub group?

I have a set E defined in ℝXℝ (E=ℝXℝ) and the operation * defined like this ...
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Issues with a particular bilinear form and determining rank, signature, etc. of its restriction

Let $b: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$ such that $b(X,Y)=trace(X^tAY)$, where $X^t$ is the transpose of $X$ and $A=\begin{pmatrix} 2 & 1\\1 & 0\end{pmatrix}$. In my ...
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1answer
19 views

Derivative over scalar field with respect to fixed point proof.

Prove there is no such scalar field that $f '(a;y) >0$ for fixed point $a$ and every non-zero vector $y$. I posted this question but some of you pointed out that it is not clear. So, $f ' (a;y)$ ...
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2answers
29 views

Proving linear dependency for two vector groups

The question: Let V be a vector space over $\mathbb{R}$. Let $S = \{v,u,w\}$ be a group of 3 vectors in V. Let T be defined as $T = \{v, v + u, v + u + 2w \}$. Prove that if S is linearly dependent, ...
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33 views

Eigenvector Problem

Given a matrix $X$, let $eigvec(X)$ be its eigenvector associated with the largest eigenvalue. Is there a relationship among $eigvec(X+X^T)$, $eigvec(X)$ and $eigvec(X^T)$? In other words, can I use ...
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188 views

Invertibility of a Kronecker Product

Prove that $A\otimes B$ is invertible if and only if $B\otimes A$ is invertible. I don't have a clue where to start to be honest. I am not very familiar yet to the Kronecker Product so could you ...
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1answer
28 views

Finding the corresponding Perron eigenvalue

Find the Perron root and the corresponding Perron eigenvector of A. $\begin{bmatrix} 0 &1 &1 \\ 1&0&1 \\ 1&1&0 \end{bmatrix}$ I figured out the Perron root which happens to ...
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1answer
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nullity and rank of the linear transformation $T: T [ p (x)]= p(x+1)$

Let $V$ be the linear space of all polynomials $p(x)$ of degree $\le n$. if $p$ belongs to $V$ and $q = T(p)$, means that $q(x) = p(x+1)$ for all real $x$. find nullity and rank of the linear ...
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2answers
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Find the matrix $M$, given four vectors

If you have $4$ vectors in a plane $x_1, x_2, b_1, b_2$, and a matrix $M$ such that $Mx_1 = b_1$ and $Mx_2 = b_2$, how do you find $M$ from this given data? Any hints would be appreciated; I am not ...
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310 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: \[ g(1)=\alpha \] \[ g(2n+j)=3g(n)+\gamma n+\beta_j \] \[ \mbox{for}\ j=0,1\ \mbox{and}\ n\geq1 \] I ...
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Give me an idea [on hold]

We've got an infinite number of cards, each of them having a positive integer written on it.Prove that however we choose 2015 cards, having the sum of the numbers written on them 4028, we can divide ...
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1answer
31 views

quotients and direct sums

Let $H$, $K$, $W$, be submodules of a module $M$. Is it true that $(H \oplus K)/W \cong H/W \oplus K \cong H \oplus K/W$? The first seems to follow from 1st isomorphism theorem on the map $\phi = ...
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1answer
29 views

Nullity and rank of the linear transformation $T[f(t)] = \int_a^b f(t) \sin (x-t) dt ~\forall~x \in [a,b]$

Let $V$ be the linear space of all real functions continuous on $[a, b]$. If $f\in V, g=T(f)$ means that $$g(x)=\int_a^b f(t)\sin(x-t)\,dt\hspace{1 cm} for\ a\le x\le b$$ Then, the nullity and rank ...
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25 views

Algebraic multiplicity of an eigen value

Let $T$ be an operator on a complex Vector space $V$. Then, the algebraic multiplicity of an eigen value is equal to $\dim ~null~ (T - \lambda I)^{\dim V}$ Which means, if we obtain the upper ...
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1answer
30 views

How to calculate time-of-flight and target hit point of a ball thrown against a wall?

Imagine you are throwing a ball against a distant wall, the question is how to find the time taken by the ball to reach the wall and also the point of impact on the wall (after the ball has bounced ...
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1answer
19 views

The group action of $S_n$ given a partition of $n$

We know that irreducible representations of $S_n$ are given by partitions of $n$. I would like to know if there is a way to explicitly write down the action of some $g \in S_n$ on the representation ...
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23 views

Linear maps and subspaces

The set-up for my question is this, let $k \le n$, let $E \subseteq \mathbf{R}^n$ be a $k$-dimensional subspace. Let $I \subseteq \{1,\ldots, n\}$ such that $|I| = k$, then we can define coordinate ...
2
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1answer
23 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
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1answer
18 views

Adjoint Transformations and Self-Adjoint Operators

I don't quite understand the whole adjoint and self adjoint thing. I know their definitions: Given a linear transformation $A:\mathbb{R}^d \to \mathbb{R}^m$, its adjoint >transformation, ...
3
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4answers
177 views

Identify a formula for each entry of the matrix $\small \begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix}^n$

Identify a formula for each entry of the matrix $\begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix}^n$. It's easy to find a solution by just looking at the first few results: \begin{pmatrix} ...