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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Inverse matrix proof

Let $A$ be a $n \times n$ matrix. Show that if $A^2=O$ then $A$ is singular, but $I−A$ is nonsingular and $(I−A)^{-1}=I+A$. What I have tiedy: $(I-A)*(I+A)=I-A+A-A^2$ $=I-A^2$ $=I-0$ since A^2=0 ...
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36 views

f is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^−1 (V ))$ = V prove?

$f$ is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^{−1} (V ))$ = $V$ This is an assertion and i said it was true. But i am confused as to what is referred to as the domain and range in this question. I would say ...
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2k views

Solutions of homogeneous linear differential equation form a vector space

Show that the solutions of a homogeneous linear differential equation $y''+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension? I understand that the dimension is 2 and that 0 is a ...
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3answers
313 views

Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
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Skew Symmetric Matrix Properties

We have a theorem says that "ODD-SIZED SKEW-SYMMETRIC MATRICES ARE SINGULAR" . Proof link is given here if needed. Now let us assume we have a $3\times 3$ skew symmetric matrices of the form $ ...
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3answers
47 views

Product symmetry matrix proof

Let $A$ and $B$ be symmetric $n×n$ matrices.Prove that $AB = BA$ if and only if $AB$ is also symmetric. So we need to prove that AB is symmetric. This means $(AB)^T=AB$. Recall a property of ...
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96 views

Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
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234 views

Diagonalizing $xyz$

The quadratic form $g(x,y) = xy$ can be diagonalized by the change of variables $x = (u + v)$ and $y = (u - v)$ . However, it seems unlikely that the cubic form $f(x,y,z) = xyz$, can be ...
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2answers
356 views

matrix singular proof

Let A, B be n×n matrices. Show that if AB = A and B≠I then A must be singular. I was thikning to prove it by contradiction, showing if A is nonsingular then we have thta AB=BA=A, therefore B is the ...
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171 views

How to efficiently solve a series of similar matrix equations using the LU decomposition

This is the problem I'm dealing with: Let $\sigma_1,\dots,\sigma_n \in \mathbb{R}$ and $b_1,\dots,b_n$ be column vectors of length $n$. Consider the system $$ (A - \sigma_jI)x_j = b_j, \quad ...
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42 views

Subsets that are also vector spaces

The vector space $R^3$ and the subset M consists of the vectors $(\xi_1,\xi_2,\xi_3)$ for which i) $\xi_1 = 0 $ ii) $\xi_1 = 0$ or $\xi_2 = 0 $ iii) $\xi_1 + \xi_2 = 0 $ iv) $\xi_1 + \xi_2 = 1 $ ...
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0answers
35 views

Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
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2answers
51 views

Is my answer correct?

I'm trying to solve this question: My solution: Since $\varphi$ is continuous we have: $C\text{ is convex}\implies C\text{ is connected}\implies \varphi(C)\text{ is connected}\implies \varphi(C) ...
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0answers
492 views

Quaternions: Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship ...
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2answers
33 views

Show that B is singular

This is a linear algebra problem concerning singularity and linear independence. A is an $n \times n-1$ matrix where $A=\{A_1,A_2,...,A_{n-1}\}$ Show that $B$ is singular if ...
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1answer
56 views

Inverse rotation transformations

I'm taking the 2-degree gibmle system and position its alignment point in a arbitrary position (denoted by the axes angles phi for the first degree, and theta for the second). How can I reverse the ...
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28 views

Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...
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84 views

Multiplication of Rotation Matrices in quaternion

Given Data and specifications NB : * means multiplication Suppose we need to rotate a point $P = \begin{pmatrix} x\\ y\\ z \end{pmatrix}$ with rotation matrix ${Q}_{3\times3}$ then what we do is ...
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1answer
79 views

For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
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106 views

Decomposition into generalized eigenspaces

I would be very grateful if someone would check the following proof for me. I came up with it as an alternative to the longer proof in the book I am reading. Many thanks! Theorem. Let $V\neq\{0\}$ ...
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2answers
91 views

$|b-a|=|b-c|+|c-a| \implies c\in [a,b]$

We know that if $c\in [a,b]$ we have $|b-a|=|b-c|+|c-a|$. I'm trying to prove that if the norm is induced by an inner product, then the converse holds. I need a hint or something. Thanks in advance
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5answers
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Equation of the line passing through the intersection of two lines and is parallel to another line.

The Question is : Find the equation of the line through the intersection of the lines $3x+2y−8=0,5x−11y+1=0$ and parallel to the line $6x+13y=25$ Here is how I did it.. $L_1 = 3x + 2y -8 = 0$ $L_2 ...
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2answers
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Characterization of definite positive matrices

We can define a positive definite matrix $A\in M(n\times n)$ as the symmetric matrix where $X^tAX\gt 0$ for every column vector $X\ne 0$ in $n$ coordinates. Suppose $A$ is symmetric, I would like to ...
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513 views

Prove that row rank of a matrix equals column rank

Let $A \in \mathbb{F}^{m \times n}$. How do you prove that row rank of a matrix equals column rank ? This question has been addressed here and here, but the explanation in one case was descriptive ...
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1answer
74 views

Necessary and sufficient conditions to have an inner product in $\mathbb R^2$

I'm trying to solve this question: Given real numbers $a, b, c$, in order to exist an inner product in $\mathbb R^2$ such that $\langle e_1,e_1\rangle=a$, $\langle e_1,e_2\rangle=\langle ...
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2answers
45 views

Linear mapping between vector spaces.

I'm curious to see if the following mapping is in fact bijective. Let $P(\mathbb{R})$ be the space of all polynomials with real coefficients. Let $f\in P(\mathbb{R})$. Then is $f(x)\mapsto ...
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1answer
26 views

parallelepiped volume with a variable

Giving this three vectors : $$ \vec{a} = \vec{i} + \vec{j} - \vec{k}$$$$\vec{b}=2\vec{i}+\vec{j}-\vec{k}$$ $$\vec{c} = m\vec{i} - \vec{j} + m\vec{k} $$ What value must have $m$, if the volume of the ...
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45 views

determinant in terms of quadratic form evaluated at a point

Say $A$ is a $n$ by $n$ positive definite matrix. Let $b$ be a column vector in $\mathbb{R}^n$. Consider the following quantity: $$b^TA^*b$$ where $A^*$ is the cofactor matrix of $A$. A simple ...
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Show that if matrix $A$ is symmetric, then so is $P^TAP$.

I need to show that if $A$ is symmetric, then so is $P^TAP$, assuming the matrix multiplications are valid. I'm sure if I actually expanded the matrices to show the entries and did the ...
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1answer
113 views

Expressing the determinant in terms of the trace of a matrix and the trace of its square

How can I prove that $$\det(A) = \frac{ 1 }{ 2 } \begin{vmatrix}\operatorname{tr}(A) & 1 \\ \operatorname{tr}(A^{2}) & \operatorname{tr}(A)\end{vmatrix}$$ where vertical bars mean the ...
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34 views

Linear Algebra Vector Tracing

Let $A(2,-1,1)$, $B$ and $C$ be the vertices of a triangle where $\overrightarrow{AB}$ is parallel to $\vec{v}=(2,0,-1), $$\overrightarrow{BC}$ is parallel to $\vec{w}=(1,-1,1)$ and $\angle(BAC)=90°$. ...
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1answer
115 views

Relation between Riccati Algebraic Equation and optimization problem

Reading this page: http://www.mathworks.com/help/robust/ug/minimizing-linear-objectives-under-lmi-constraints.html I got stuck in the result that says it can be show that minimizing Trace of X (a ...
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1answer
43 views

Let $Q$ be a symmetric $n$ by $n$ matrix, there exists an orthogonal matrix $F$ such that $F^TQF=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$

Let $Q$ be a symmetric $n$ by $n$ square matrix, there exists an orthogonal matrix $F$ such that $$F^TQF=\operatorname{diag}(\lambda_1,\ldots,\lambda_n),$$ with $\lambda_1,\ldots,\lambda_n$ being ...
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199 views

Find the eigenvalues and eigenvectors of the linear transformation $T(x,y,z)=(x+y,x-y,x+z)$. Verify that the eigenvectors are orthogonal.

Find the eigenvalues and eigenvectors of the linear transformation $T(x,y,z)=(x+y,x-y,x+z)$. Verify that the eigenvectors are orthogonal. Part A: $$T(x,y,z)=\begin{pmatrix} 1 & 1 & 0 \\ 1 ...
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Linear transformation in linear algebra

Let $e_1= \begin{bmatrix} 1\\ 0 \end{bmatrix} $ Let $e_2= \begin{bmatrix} 0\\ 1 \end{bmatrix} $ Let $y_1= \begin{bmatrix} 2\\ 5 \end{bmatrix} $ $y_2= \begin{bmatrix} -1\\ 6 \end{bmatrix} $ Let ...
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1answer
49 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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1answer
265 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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49 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
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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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387 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 ...
5
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1answer
221 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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6answers
121 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.
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2answers
72 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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32 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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0answers
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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 ...
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3answers
174 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
61 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
114 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
75 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 ...