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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Linear Algebra Vectors Using Planes and Lines to Find Coordinate

Let A(2,-1,1)), B and C be the vertices of a triangle where ($\overrightarrow{AB}$) is parallel to ([2,0,-1]), ($\overrightarrow{BC}$) is parallel to ([1,-1,1]) and the angle at (A) is a right angle. ...
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laplace transform of sine with modulous

L{|sin wt|}= Iam typing this because it asked for thirty characters don't mind and this may seem silly and hence i tried to solve it using the relation used for the formula used to find the laplace ...
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1answer
21 views

Prove that if $T: V \to W$ is one to one and ${Tv_1, … Tv_n}$ is a basis for W, then ${v_1,…, v_n}$ is also a basis for V.

Prove that if $T: V \to W$ is one to one and ${Tv_1, ... Tv_n}$ is a basis for W, then ${v_1,..., v_n}$ is also a basis for V. My idea is to introduce a $T^{-1}$ and then do a proof that is similar ...
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25 views

For what value of K is the following identity

This is quite a difficult equation to solve for me. Where is the best way to start with?
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15 views

Semi direct product

Prove that (i) $GL_n(R)= \coprod_{w\in S_n} UwB$ where $w \in S_n$ is a permutation matrix. and $U$ is a subgroup of $GL_n(R)$ consisting of upper triangular matrices with diagonal entries $1$ and ...
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1answer
55 views

Find $B$ if $AB=BC$ and $A,C$ are invertible

Suppose $A$ and $C$ are known invertible complex matrices of possibly different orders. If $B$ is an unknown matrix of appropriate order such that $AB = BC$, then how could one solve for $B$?
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1answer
33 views

Using linear algebra, analyze the given electrical circuits by finding the unknown currents

Problem Using linear algebra, analyze the given electrical circuits by finding the unknown currents. Progress I have these 6 equations: $10=20I_5-20I_3+20I_6$ $10=20I_4+20I_5+20I_6-20I_2$ ...
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1answer
23 views

Algebra - proof verification involving permutation matrices

Theorem. Let $\textbf{P}$ be a permutation matrix corresponding to the permutation $\rho:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$. Then $\textbf{P}^t=\textbf{P}^{-1}.$ Proof. First note the following ...
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Find the dimension of a subspace by find a basis for the null space.

Below is the question and my proposed answer. It seems like it is a trick question, but maybe my answer is good enough or maybe I am wrong. Any help would be great. 2) Show that the dimension of the ...
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1answer
22 views

Verify that $(I−XY)^{(-1)}*X=X*(I−YX)^{(-1)}$ [duplicate]

Verify that $(I_n−XY)^{-1}\cdot X=X\cdot (I_m−YX)^{-1}$ The first $I$ is of order $n$ and the second is of order $m$. $X$ is $n\times m$ $Y$ is $m\times n$
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22 views

For what values of $k$ will these equations have no solution/infinite solutions/unique solution

Here are the 3 linear equations: $$x+y-z=-1$$ $$2x-4y-6z=-1$$ $$x-y+(k^2-1)z=k$$ I understand a $4\times3$ matrix must be set up in order to solve this particular problem.The part which I get ...
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0answers
20 views

Easy elementary proof of Farkas Lemma?

Is there any elementary proof of Farkas lemma which does not use convex analysis and hyperplane separation theorem? What about special case below: If the Matrix $A$ is invertible, then there is ...
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5answers
51 views

If $m$ is the smallest positive integer such that $T^m = 0_v$, then $m \leq \dim(V)$

Let $V$ be a vector space and $T \in L(v)$. Prove that If $m$ is the smallest positive integer such that $T^m = 0_v$, then $m \leq \dim(V)$ I have no idea how to prove this.
2
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2answers
24 views

Kernel of a Linear Map on A Tensor Product

Suppose I have the linear maps $ l,k: V \otimes V \rightarrow V \otimes V$ defined by $ l( e_{i_1} \otimes e_{i_2} ) = e_{i_1} \otimes e_{i_2} + e_{i_2} \otimes e_{i_1}$ and $ k( e_{i_1} \otimes ...
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1answer
17 views

Explain this “cross-multiplication”

I am working out a text book problem, in one of the steps the author takes two linear equations with 3 variables each... $$ l(2)+m(3)+n(1)=0\tag{1} $$ $$ l(1)+m(2)+n(-1)=0\tag{2} $$ ...and arrives ...
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54 views

Determinants, how do they emerge and why? [duplicate]

This is a very simple and basic question, what's the simplest proof that the determinant is the same no matter what basis you choose? Also, I've been wondering what exactly is the determinant, what ...
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25 views

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 ...
2
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3answers
144 views

Inner product is a function from…to…?

Example, if $v,w \in \mathbb{R}^2$, then the inner (dot) product defined by $$f(v,w) = \left< v,w \right>$$ is bilinear, so is $f$ a function from $\mathbb{R}^2 \times \mathbb{R}^2 \to ...
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2answers
32 views

how to determine zero entries in a vector

I am writing an optimization expression and in the constraints part, I want to limit the number of non-zero entries of the vector to a certain number R. Suppose if the vector is M dimensional, then ...
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1answer
40 views

Canonical Isomorphism between coker of the dual and ker

Given $L$ and $M$ be finite-dimensional vector spaces and let $g:L\to M$ be a linear map. Then there exists a canonical isomorphism from $\operatorname{coker} g^*$ to which of the following spaces - ...
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4answers
31 views

optimization of coefficients with constant sum of inverses

Does anybody knows if there is an easy solution to the following problem: Given $A = [a_1, a_2, ... a_n]$ and K, find B = $[b_1, b_2,...b_n]$ that minimizes $AB^T$ such that ...
3
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1answer
19 views

Show Moore-Penrose Inverse is equivalent to standard inverse if A is invertible and nonsingular

Show Moore-Penrose Inverse $A^+$=$A^-$ if A is invertible and nonsingular: I want to check that I doing this proof correctly. Using the 4 properties of the Moore-Penrose Inverse, I believe I show it ...
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1answer
141 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
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1answer
35 views

Multiplicity as roots of the minimal polynomial

Let $V\neq\{0\}$ be a finite-dimensional vector space over a field $F$ and let $\alpha \in \text{End}(V)$. Suppose that $\lambda$ is an eigenvalue of $\alpha$ with multiplicity $r$ as a root of the ...
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37 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
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33 views

Can't understand solution to Linear Algebra problem.

http://www.cms.zju.edu.cn/UploadFiles/AttachFiles/201082333637670.pdf#page=81 I do not understand the solution to (b) (problem is on the current page, solution starts next page). The solution says ...
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0answers
38 views

P-norm Unit Ball

Proof that for $0<p<1$, $p\in \Bbb{R}$ $$\|(x,y)\|_p=(|x|^p+|y|^p)^{\frac{1}{p}}$$ doesn't define a norm in $\Bbb{R}^2$. However, $$d_p((x_1,x_2),(y_1,y_2))=\sum_{i=1}^2|x_i-y_i|^p$$ defines a ...
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41 views

Battle Ship Winning Algorithm - Optimal Strategy

I have an $8 \times 8$ grid. I have three ships that are $4$ long, $3$ long, and $2$ long. Is there an algorithm that can ensure a win every time? Oh! Most importantly, you must know the number of ...
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2answers
30 views

How to tell if a columns of matrix are linear dependent?

How can it be seen if the following matrix is linear dependent? Let $A= \begin{bmatrix} 0 & -3 & 9& \\ 2&1& 7 \\ -1& 4 &-5 \\ 1&-4&-2 \end{bmatrix} $ First ...
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1answer
19 views

Prove transpose of pseudoinverse commutes

How can I show that $(A^T)^+=(A^+)^T$, where $A^+$ is Moore-Penrose Inverse? I know there are 4 properties of the Moore-Penrose Generalized inverse, for example: $$AA^+A=A^+. $$ To prove it, could I ...
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Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
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Prove that the dual space $V^{\ast}$ has the direct-sum decomposition $V^{\ast}=V_1^0\oplus \cdots \oplus V_k^0$.

Let V be a vector space, let $W_1, \ldots ,W_k$ be subspaces of $V$, and let $V_j= W_1+ \cdots+W_{j-1}+W_{j+1}+ \ldots + W_{k}$. Suppose that $V=W_1\oplus \cdots \oplus W_k$. Prove that the dual ...
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3answers
58 views

$T$ be linear operator on $V$ by $T(B)=AB-BA$. Prove that if A is a nilpotent matrix, then $T$ is a nilpotent operator.

Let $V$ be a vector space of $n\times n$ matrices over a field F, and let $A$ be a fixed $n\times n$ matrix. $T$ be linear operator on $V$ by $T(B)=AB-BA$. Prove that if A is a nilpotent matrix, ...
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1answer
63 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
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4answers
35 views

commutative matrix multiplication of nxn matrices?

If there are two matrices A and B that are both nxn matrices, will AB = BA always? Is there a way to have those two matrices so that AB = 0 but BA ≠ 0?
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Does this formula take constant value?

Now, $x_i, \xi, f \in R^n(i= 1, 2, \cdots , k)$, and \begin{align} \sum_{i=1}^k x_ix_i^T\xi=f \end{align} holds. If the above equation is solvable about $\xi$, the value of $f^T\xi$ doesn't depend on ...
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3answers
66 views

Do I justify it well?

I have a function that is of the form and I want to prove that it is always positive : $$\sqrt{x^{4}-7x^{2}+16}$$ I say that I can study $x^{4}-7x^{2}+16$ by putting $X = x^{2}$, which gives me ...
2
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1answer
50 views

For all $n$, is there a real $n\times n$ matrix that can't be written as the sum of two commuting squares?

My original problem was to prove that an even degree real polynomial which acts as a function from the set of real $n\times n$ matrices to itself cannot ever be surjective. Now, I can negate the ...
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4answers
43 views

Finding the diagonalizing matrix.

Find a nonsingular matrix $C$ such that $C^{-1}AC$ is a diagonal matrix. $$ A=\begin{pmatrix} 1 & 0 \\ 1 & 3 \\ \end{pmatrix} $$ I have found the eigenvalues to be 1 ...
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A Difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
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How to justify that a basic feasible solution to a Linear Program corresponds to an extreme point of the feasible region?

Say we have an LP Problem in standard form. That is, $$\text{Maximise} \;\; C^T X $$ $$ \text{subject to:} \;\;\; AX = B --(1) \;\;\;\; \text{where $A$ is an $m \times n$ matrix }$$ I read ...
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704 views

Are 10x10 matrices spanned by powers of a single matrix?

I don't know how to answer this question: Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the ...
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0answers
17 views

Discretization of a continuous time-invariant linear system

I have the following autonomous system $$\dot{x}(t) = Ax(t)$$ where $x \in \mathbb{R}^2$ and $A$ is a constant matrix with suitable dimensions. When I discretize this system under a sampling time of ...
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1answer
36 views

Question about dimension of a subspace

Let $K$ be a field and define the following subspaces $$V=\textrm{span}(e_1,e_2,e_3),\;\; V^\bot = \textrm{span}(e_4,e_5,e_6)$$ inside $K^6$. Let $\dim L=4$ and assume that $\dim L\cap V\leq 1$. Can ...
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1answer
20 views

Find an $\alpha$ such that $T_i\alpha \neq T_j\alpha$

Suppose that $T_1, \cdots , T_m$ are linear operators from linear space $V$ to $V$, such that $T_i \neq T_j$ for all $i \neq j$. Prove that there exists an $\alpha \in V$ such that $T_i\alpha \neq ...
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2answers
49 views

Prove that an element of the basis is an element of the Kernel after linear transformation

Let $T:R^4\rightarrow R^4$ and basis $B=(v_1,v_2,v_3,v_4)$. $$T(v_1)+T(v_2)=T(v_3)\; \text{ and } \; T(v_1)+T(v_3)=T(v_2)$$ Prove that $v_1\in Ker(T)$ What I wrote is: $$T(v_1)=T(v_3)-T(v_2)\; ...
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1answer
21 views

LQ decomposition and inequalities

Suppose I have an element-wise inequality: $Ax \ge b$, where $A$ is a rectangular matrix with full row rank, and $x$ and $b$ are appropriately sized column vectors. I need to check if the inequality ...
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1answer
49 views

Understanding the term “Abstraction” in mathematics

When the need for abstraction is asserted in mathematics is it generally meant that there is a need to apply a definition to n-dimensions such that n is an integer going to infinity?
3
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1answer
46 views

Matrix manipulation using trace

Suppose that $u$ is an $N\times 1$ random vector and $M$ is an $N\times N$ nonrandom positive semi-definite matrix that is also idempotent: $M\times M=M$. Claim: $E(u'Muu'Mu)=\text{Tr}\{M ...
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1answer
16 views

Show that Two Vectors Making Supplementary Angles?

I just need a start. I am not looking for whole prove, but it'd be more appreciated if I get one. Q. Use Theorem u . v = |u| |v| cos a and the trigonometric identity, cos (180-a) = -cos a, to ...