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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Issues with connecting the SVD and Eigenvalues for block matrix

In class, we have talked about the singular value decomposition and its connection to Eigenvalues. Specifically, for a matrix A, if the columns of a matrix contain linearly independent eigenvectors, ...
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Collinear and Linear Dependence

Prove that a set of 3 points a, b and c in the plane are collinear if and only if the vectors [a, 1], [b, 1], [c, 1] are linearly dependent, i.e. one of these points can be written as a linear ...
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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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Matrix Representation of a linear operator and Checking if it is surjective and injective

Let $W_1$ be the subspace of C(0,1) spanned by the functions $\{e^x,xe^x,x^2e^x\}$. Let $W_2$ be the subspace of C(0,1) spanned by the functions $\{1,e^x,xe^x,x^2e^x\}$. Let T be the application ...
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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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Does the binary operation $m ⋆ n = m^n$ on N have a neutral element?

Does the binary operation $m ⋆ n = m^n$ on N have a neutral element? I said yes and it is 1 such that $m ⋆ e = m^e = m$ but apparently that is wrong
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2answers
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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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How to proof the cofactors expansion for determinants?

I have a homework, where I must prove that the cofactors expansion for determinant is equal to the permutations definition of it. I have been thinking about it and my first idea is by induction in the ...
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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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1answer
22 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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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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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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18 views

Quaternions- Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represnts the position vector as result of rotation with an angular veclocity $\omega(t)$ in quaternion , then you can make the relationship ...
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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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Question about eigenvalue problem of a selfadjoint operator.

Let $x=(x_1,x_2)$, and let $X_m$ denote the space of homogeneous polynomial vector fields on $\mathbb{R}^2$ of degree $m$. For example if $m=2$ a vector field $U\in X_2$ is of the form $$ ...
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1answer
8 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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11 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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1answer
19 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
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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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27 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
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$|b-a|=|b-c|+|c-a| \implies c\in [a,b]$

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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4answers
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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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Equation of the line passing through the intersection of two other lines. [on hold]

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 $6x+13y=25$
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matrices and eigen values [NBHM-2014]

In each of the following cases, describe the smallest subset of $\Bbb{C}$ which contains all the eigenvalues of every member of the set $S$. a. $S = \{A ∈ M_n(\Bbb{C}) | A = BB^*, B \in ...
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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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19 views

This really basic thing true? (Yes or no question)

These days, I'm reviewing linear algebra. Here is a theorem in my text Let $T:V\rightarrow W$ be a linear transformation If $\beta$ is a basic for $V$, then Span$(T(\beta))=im(T)$. ...
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1answer
15 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
29 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
30 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
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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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1answer
19 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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2answers
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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
51 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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1answer
22 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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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
23 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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Prove that if A and B are both invertible nxn matrices then A and B are row equivalent to each other.

Prove that if A and B are both invertible nxn matrices then A and B are row equivalent to each other. I dont see how this is even remotely true, I know that an invertible matrix can be reduced to I, ...
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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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2answers
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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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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
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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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27 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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28 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
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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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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
27 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
23 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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26 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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24 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 ...