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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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 ...
3
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0answers
55 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 ...
2
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0answers
28 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
145 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
41 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 ...
0
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1answer
44 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
37 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
20 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 ...
1
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1answer
146 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 - ...
2
votes
1answer
36 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 ...
4
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0answers
41 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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0answers
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 ...
0
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0answers
42 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 ...
1
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1answer
52 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
35 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
21 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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0answers
61 views

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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0answers
11 views

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 ...
4
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3answers
61 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, ...
0
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1answer
64 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
37 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 ...
3
votes
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
votes
1answer
53 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 ...
0
votes
4answers
44 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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0answers
17 views

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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0answers
9 views

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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2answers
898 views

Are $10\times 10$ 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
21 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 ...
0
votes
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 ...
3
votes
2answers
52 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)\; ...
1
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1answer
24 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
50 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
47 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
17 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 ...
0
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1answer
20 views

Rearrange equation with integrating factor

I'm trying to do the following in the middle of a huge question involving a differential equation - I need to rearrange this equation for t, but have no idea where to start. First image is the ...
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1answer
31 views

proving a matrix is symmetric to its squared version

Prove: If $A^T∗A=A$, then $A$ is symmetric and $A=A^2$. I have, $A=A^T∗A=A∗A=A^2$. I believe it is too short. Any help?
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23 views

Elementary Linear Algebra - Linear system with various conditions

Under what conditions of $a$ and $b$ will the following linear system have no solutions, one solution, or infinitely many solutions? $$2x - 3y = a$$ $$4x - 6y = b$$
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1answer
30 views

proving a matrix is symmetric

Let $A$ be an $nxn $ symmetric matrix. a) Show that $A^2 $ is symmetric. b) Show that $2A^2 -3A + I$ is symmetric. for part a), i have: $A=A^T$ $A^2 = A\times A$ $A^2 = (A^T)\times(A^T)$ $A^2 = ...
4
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1answer
37 views

Are primitive row stochastic matrices diagonalizable?

Let $A$ be an $n \times n$ matrix with real, non-negative entries. Assume $A$ is primitive, meaning there exists an integer $k$ such that $A^k>0$ (here the inequality means all entries in $A$ are ...
0
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1answer
20 views

$A$ and $B$ conjugacy

Show that the matrices $A=\begin{pmatrix}2&0\\0&0\end{pmatrix}$ and $B=\begin{pmatrix}2&0\\1&0\end{pmatrix}$ are not $\mathbb{Z}$ conjugate (there exists no matrix ...
3
votes
1answer
95 views

Does every theorem in $\mathbb{C}^n$ hold for $K^n$

There any many books that cover many of the topics in $\mathbb{R}^n$ and $\mathbb{C}^n$. There are many theorem in these topics. I know I cannot use $\mathbb{R}^n$ theorem on $\mathbb{C}^n$ and I ...
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1answer
27 views

Square root of a matrix proof

Let $B$ be a real symmetric $2 \times2$ matrix which satisfies: $$\sqrt{B}v_1=\lambda_1v_1$$ $$\sqrt{B}v_2=\lambda_2v_2,$$ where $v_1,v_2$ are eigenvectors of matrix $B$ and ...
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1answer
24 views

Show that a matrix A may have all leading principal minors greater or equal to zero, yet not be positive semi-definite.

Title says it all, but I'll rephrase it to be clear. A is an $n\times n$ matrix whose leading principal minors are all greater than or equal to zero. A leading principal minor is the determinant ...
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1answer
17 views

Derivates and Limits in the Same Problem are an Issue.

I am working on the following problem:- Evaluate lim x→1 [( x^1/4 - 1 ) / ( x^1/3 - 1 )] by relating it to the derivatives of functions. Now this is quite a ...
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1answer
18 views

Show that if $(QA)x = 0$ has just the trivial solution, then $A$ is invertible

Let $Q$ be an invertible $n\times n$ matrix, $A$ an $n\times n$ matrix and $x$ is an $n\times 1$ column vector (or matrix) so that the matrix equation $Ax = 0$ represents a homogeneous system of $n$ ...
0
votes
1answer
14 views

Quadratic forms — rank of matrix

Assume that $M$ is the matrix of some quadratic form (over any field of characteristic not $2$) and set $$Q(\overline{x})=\overline{x}^tM\overline{x}$$ We can replace $M$ by the symmetric matrix ...
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1answer
24 views

If we know nullspace of matrix, how to find reduced row echelon form of that matrix?

vectors u = [4 1 0 0] and v = [1 0 2 1] form a base of nullspace of matrix $$ A\in M_{5,4}(R) $$ Find a reduced row echelon form of Matrix A. Since $ n-r = dimN(A) $ we know we got two base ...