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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174 views

Abstract algebra true or false answer check

Sorry about the giant picture file, but typing up this many questions on Latex would take forever. My attempts are below, I am fairly sure 16+ are right My answers: -1T- -2T- -3F- -4F- -5T- -6F- ...
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23 views

Find an optimal subspace

Assume I have two vectors m1 and m2 on a sphere in 3-d space. I want to find a 2-d subspace that gives the highest angle (only consider 0 to 90 degree) between m1 and m2 after projecting onto some 2-d ...
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1answer
44 views

Linear algebra, matrix in $\mathbb{C}^n$ [duplicate]

, Can you help me with the following exercise? Show that doesn't exists matrix $A,B\in M_n(\mathbb{C})$ such that $$AB-BA=I$$ Has something to do with the annihilator? Thanks !
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1answer
99 views

Linear transformation matrix representation with differentiation answer confirmation

I hope you liked the title. I have a question that is as follows: Consider the linear transformation $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ given by $$T(f(x))=f(0)+f'(x)+f''(x)$$ Where the ...
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36 views

What can be said about the terminal point of the following vector if its initial point is at the origin?

In relation to the points P1 and P2 in the figure, what can you say about the terminal point of the following vector if its initial point is at the origin? I honestly have no idea what this ...
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115 views

Find scalars $a$ and $b$ so that $au + bv = (1, −4, 9, 18)$

Alright so I am running out of ideas about how to approach this problem, I tried setting $(1,-1,3,5)$ to $w$ and solving for $a$ and $b$ algebraically but just ended up with $a = (w-bv)/u$ and ...
4
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1answer
106 views

Finding a basis for $\ker(T)$

I have this question: Let $Z\in M_{2\times2}(\mathbb{R})$ be defined as $$Z = \left( \begin{align} 1 &&1\\1 &&1 \end{align} \right)$$ and consider $T: ...
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86 views

how do i tell if a vector is parallel to another vector in R^6?

So far in my book I haven't learned any of the parallel or perpendicular notation.. so there must be some way to tell this answer that the book hasn't told me.. I looked back and there was nothing ...
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1answer
30 views

Show that some endomorphsm is not diagonalizable

Given an endomorphism $f:V \rightarrow V$ on an $\mathbb{R}$-vector space, prove that if there is $v \in V-\{0\}$ such that $f^2(v)=-v$, then $f$ is not diagonalizable.
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34 views

Show that $\DeclareMathOperator{im}{Im} \im(\alpha) \cap \im(\beta)={0_v}=\ker(\alpha) \cap \ker(\beta)$

I am just completely stuck on this problem, it may just be me confusing the vocabulary and what they mean Question: Let V be a finite-dimensional vector space over a field F and let $\alpha, \beta$ ...
4
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2answers
65 views

Basis and dim of the set of all $n\times n$ symmetric matrices.

An $n \times n$ square matrix $A$ is called symmetric if $A^T = A$ Show that the set of all $n \times n$ symmetric matrices, denoted $S$, is a subspace of $M_n(\mathbb{R})$. Give a basis for $S$ ...
11
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272 views

Are these parallel theorems from Set Theory and Linear Algebra connected through Category Theory?

From Set Theory and Linear Algebra, we have these two theorems: Given two finite sets of the same cardinality $X$ and $Y$ and a function $f:X\rightarrow Y$, the following are equivalent: $f$ is a ...
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1answer
114 views

Prove that $Im(A)+Ker(A)=R^n \iff Ker(A^2)=Ker(A)$

$\def\Im{\operatorname{Im}}\def\Ker{\operatorname{Ker}}$How to prove that for any squared matrix such that $ \Im(A)+\Ker(A)=\mathbb{R}^n$ if and only if $\Ker(A^2)=\Ker(A)$. It is evident to me that ...
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33 views

Invertibility of Matrix by examining the column vectors

Here is my question: Give a matrix B, determine the invertibility of B by examining the column vectors of B. I have solved this one way, by trying to show that the column vectors are independent, by ...
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48 views

Is it true for algebras A,B,C, that $(A+B)\cap C = A\cap C+B\cap C$?

Let $A,B,C$ be subalgebras of some algebra $X$. I've managed to show, that $A\cap C+B\cap C\subseteq (A+B)\cap C$. If $x\in A\cap C+B\cap C$, then $x=a+b$, where $a\in A\cap C, b\in B\cap C$. Since ...
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1answer
31 views

Every choice of basis is equally natural?

Let $V$ denote a vector space of dimension $n$ over a field $\mathbb K$. Then I read: In practice, there is typically no choice of basis which seems more natural than the other choices. To convince ...
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2answers
81 views

Show ker($\alpha$)=ker($\alpha$)^2 iff ker($\alpha$) and im($\alpha$) are disjoint

Let $V$ be a vector space over a field $F$ and let $\alpha$ be an element of $\operatorname{End}(V)$. Show $\ker(\alpha)=ker(\alpha^2)$ iff $\ker(\alpha)$ and $\operatorname{im}(\alpha)$ are disjoint. ...
5
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124 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
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2answers
133 views

Question about transpose of quaternion matrix

If $A$ is a matrix with entries in the quaternions and $q$ is a quaternion is $(qA)^T = q A^T$ or $(qA)^T = \overline{q}A^T$?
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25 views

Definition of linear subspace

Let $k<d\in\mathbb{N}$. Given the following definition: $G=\{ S: S\text{ is }k\text{-dimensinal subspace of }\mathbb{R}^d\}$ Would you understand that $G$ contains only "homogeneous linear ...
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222 views

Closest points on two line segments

I am looking for a general formulation to find the closest points on two line segments. What I was thinking about is to define our lines as: $$ P1 + s (P2-P1)$$ $$ Q1 + t (Q2-Q1)$$ Where $P1 , ...
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3answers
107 views

Finding ker, im, dim of a linear transformation

1Ok, I am a student trying to wrap my head around some of these concepts and need help understanding how to approach some problems. Question: Let $\alpha:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the ...
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1answer
67 views

How to find a subspace gives maximum angle between two vectors

If there are two vectors in 3-d space, how to find a subspace gives maximum angle between two vectors after projection? Assume we only consider angles between 0-90 degree and the projection happens on ...
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2answers
118 views

Simplifying the expression of a product of inner products

Let $\mathbf{v}=(v_1,\cdots,v_n)^T, \mathbf{w}=(w_1,\cdots,w_n)^T, \mathbf{a}=(a_1,\cdots,a_n)^T \in\mathbb{R}^n$, and let $$ A = ...
2
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1answer
49 views

Base of subspaces and intersections

Let $V$ be a vectorial space with base $[v_1,v_2,v_3,v_4]$. Let $ U \subset V $ be the subspace generated by $ u_1 = v_1-v_2+v_4 $, $ u_2 = v_3+v_4 $,$ u_3 = v_1-v_2-v_3 $ and $ W \subset V$ the ...
5
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1answer
169 views

When is the equation $Ax = b$ solvable in the integers?

Let $A$ be an $m\times n$ matrix with integer entries, $b$ a column-vector with $m$ integer entries. Suppose the equation $Ax = b$ has infinitely many solutions. It is clear that the general ...
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1answer
201 views

Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
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1answer
94 views

A weird kind of dot product?

In the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf at page 499, the authors defined the dot product of two vectors in a weird way: In the 3D Euclidian space let $\textbf{A}$ and ...
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28 views

Block Circulant matrix

Let \begin{align} A=\left(\begin{array}{cc} B & C \\ -C & B \end{array}\right), \end{align} where $B=\left(\begin{array}{ccc} \alpha^R & \beta^R & \gamma^R \\ \gamma^R &\alpha^R ...
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1answer
137 views

For the matrix A as given below, which of them satisfy $A^6=I$?

For the matrix A as given below, which of them satisfy $A^6=I$? $A=\left(\begin{array}{ccc} \cos\frac{\pi}{4}&\sin\frac{\pi}{4}&0\\ -\sin\frac{\pi}{4}&\cos\frac{\pi}{4}&0\\ ...
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25 views

How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
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1answer
63 views

fast way to find the characteristic polynomial

I need to find the eigenvalues and eigenvectors of this matrix. $ \left( \begin{array}{ccc} 7/34 & -11/34 & 4/17 & -1/17 \\ -11/34 & 27/34 & 1/17 & 4/17 \\ 4/17 & 1/17 ...
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1answer
29 views

The idea of eigenvectors and a transforming matrix

I'm reading this tutorial on PCA: http://nyx-www.informatik.uni-bremen.de/664/1/smith_tr_02.pdf I quote from it: It is the nature of the transformation that the eigenvectors arise from. Imagine a ...
2
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0answers
59 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
0
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1answer
37 views

Constructing an eigenvector for a certain matrix representing a graph with a perfect code

Let $A$ be a symmetric $(0,1)$-matrix whose row sum is $r.$ Suppose I have a $(0,1)$ vector $v$ such that $$Av = \vec{1} - v.$$ By taking the vector $$u = \vec{1} - (r+1)v$$ we see that $$Au = A ...
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1answer
141 views

can someone help me to prove rank(P A) = rank(A).?

is that correct and we should use the hint but how we use it correctly??
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1answer
50 views

Circulant matrix

$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$ Here $A$ is the block circulant matrix and B and C are $n \times n$ matrices which are circulant. How can write it as in roots ...
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1answer
52 views

Given the permutation $\sigma=\left(\begin{array}{ccccc} 1&2&3&4&5\\ 3&1&2&5&4 \end{array}\right)$

Given the permutation $$\sigma=\left(\begin{array}{ccccc} 1&2&3&4&5\\ 3&1&2&5&4 \end{array}\right)$$ the matrix A is defined to be the one whose $i$-th column of the ...
4
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2answers
94 views

Determinant and trace as conjugations?

For real matrices $A$ it holds that $$\det\,\big(e^A\big)=e^{\mathrm{tr}\,A}$$ so we can write $$\mathrm{tr}=(\exp)^{-1}\circ \;\det\;\circ\;(\exp).$$ Is this interpretation of trace as the ...
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198 views

Linear algebra calculus trick.

I have a matrix and a vector: $$ A=\begin{bmatrix} a &b\\ c&d \end{bmatrix}, $$ $$ \vec v=\begin{bmatrix} a+b\\ c+d \end{bmatrix} $$ Is there an algebraic operation that produce the ...
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1answer
121 views

Geometric interpretation of Laplace's formula for determinants

Coming from the geometric point of view, the determinant of an $n \times n$-Matrix computes the volume of an parallelepiped spanned by the columns of the matrix. In context of this question, let the ...
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3answers
53 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
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73 views

Number of integer solutions to linear equation/inequality

If I am given an equation in the form of $ax+by=c$, I understand there is an algorithm to find solutions, but is there a way to find the number of all the integer solutions in a speedy method? Can ...
0
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1answer
86 views

finding the dimension and a basis of U given a single vector

I've been given a very confusing homework problem that is as follows: Let U be the set of all vectors u in $ℝ^4$ such that $2(u_1) + 3(u_3) - 2(u_4) = 0$ (i.e. U is the solution space of a given ...
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1answer
27 views

Taking product of cofactor with different row

Given a matrix $A=(a_{ij})_{n\times n}$, let $C_{i,j}$ be the cofactor in position $(i,j)$. By the determinant formula, we have $$\det A=\sum_{i=1}^n a_{i,1}C_{i,1}.$$ What about if we take a ...
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39 views

Position vectors of sphere/circle touching central one

I am trying to understand the meaning of an expression describing the "kissing" number problem. On Wiki, it states the following: Let $x_n$ be a set of $N$ $D$-dimensional position vectors of the ...
2
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1answer
83 views

Eigenvalues of polynomials of a matrix and its inverse up to summation by identity

There is a paper that I am reading and the following has been considered without proof: (Suppose $\lambda(.)$ defines the spectrum of a matrix and one can define a random variable on this spectrum say ...
3
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2answers
43 views

Show $\cos\theta=\frac12(\text{tr}(g)-1)$ with $g\in\text{SO}(3)$

How can I show that for $g\in\text{SO}(3)$ given by $\begin{pmatrix}1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta\end{pmatrix}$ the equality ...
3
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1answer
65 views

Why is $L_A$ not $\mathbb K$ linear (I can prove that it is)

Let $\mathbb K$ denote the skew field of quaternions and $A \in M^{n \times n}(\mathbb K)$ and $X\in M^{1\times n}(\mathbb K)$. Let $L_A : \mathbb K^n \to \mathbb K^n$ be defined as $L_A(X) = ...
0
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1answer
40 views

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$?

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$? For example, Can I say that $M^R_{2x2}$ is a subspace of $M^R_{2x3}$ so it can be isomorphic to $R_4[x]$ ? (because they have ...