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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Finding the matrix for a linear transformation on a vector space when the basis changes

Let B={$u_1,u_2,u_3$} as basis of Vector Space V, and Let T: V→V be the linear operator defined by, $$ [T]_B=\begin{bmatrix} -3 & 4 & 7 \\ 1 & 0 & -2 \\ ...
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+500

Weyl group, bilinear form, and character/cocharacter pairing. Many questions!

Let $G$ be a connected linear algebraic group, $T$ a maximal torus of $G$, and $\alpha$ a weight of $T$ such that $G_{\alpha} = Z_G(S)$ is not solvable, where $S = (\textrm{Ker } \alpha)^0$. I have ...
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Diagonalizability and elementary divisors

How to prove that an $n \times n$ matrix $A$ over a field $\mathbb F$ is diagonzalizable if and only if every elementary divisor of $A$ has degree $1$? I kind of know why this is true but I am not ...
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Finding Transition Probabilities using Metropolis Hastings

I want to find the $4$x$4$ Probability Transition Matrix under the temperature parameter T=2 of Metropolis Hastings. I know that, if x and y are neighbors, $p(x,y) =$ $$ f(x) = \left\{ ...
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24 views

Lower bound for norm of matrix

I have the following problem: $A$ is a positive definite, symmetric matrix. Firstly I was required to find a matrix $B$ such that $B^n = A$. I believe this to be $C(D^{\frac1n}) C'$ where C is the ...
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If some columns of $XA, A$ are equal, does it mean $XA=A$?

I'm working on a problem related to the row space $R(A)$ of a matrix $A \in K^{k \times n}$, where $k < n$. This space is invariant under a left-action of $GL(k, K)$ on the matrix $A$. Say I have ...
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Counting the isotropic points for both quadratic and hermitian forms.

Consider an octonion algebra $\mathbb{O} = \mathbb{O}_{\mathbb{F}_{q^2}}$ over a field of order $q^2$, $q = p^k$. Then we have a natural quadratic and hermitean (by this I actually mean hermitean ...
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43 views

Does $A^2 \geq B^2 > 0$ imply $ACA \geq BCB$ for square positive definite matrices?

Assume we have two $n \times n$ real nondegenerate matrices $ A^2 $ and $B^2$, such that $$ A^2 \geq B^2 > 0, $$ where "$\geq$" means positive semidefinite (Loewner) ordering. Does the following ...
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1answer
20 views

Reduction of a representation of the Symmetric Group $S_3$

I have this representation of $S_3$ obtained in the usual way $$\varrho\left(\sigma\right)e_i=e_{\sigma_i}$$. Being more explicit the representation is this one: ...
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28 views

Eigen values of a matrix depending on k

If $A = \begin{bmatrix} 2 & k \\ 0 & 1 \end{bmatrix}$. Find all values of $k$ for which A has eigenvalues 3 and -1. A has no real eigenvalues. (David Poole, Linear Algebra). The ...
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Find characteristic and minimal polynomials [duplicate]

find the characteristic and minimal polynomial of the matrix $$B=\begin{pmatrix} a & 1 & 0 & 0 \\ b & 0 & 1 & 0 \\ c & 0 & 0 & 1 \\ d & 0 & 0 & 0 ...
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Fourier function expansion for extension over a $2\pi$ period

So I am currently looking at a fourier expansion for $$f(x)=\left\{\begin{array}{ccl}\sin x &\text{ if }& x\in[0,\pi]\\0 & \text{ if } & x\in[\pi,2\pi]\end{array}\right.$$ I am ...
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Basis of sum and intersection of two subspaces.

I have two vector subspaces: $U=span ((1,1,0,-1), (1,2,3,0), (-1,-2,-3,0)) \\ V=span ((1,2,2,-2), (2,3,2,-3), (1,3,4,-3))$ Now i have to find $B_{U+V}$ and $B_{U\cap V}$ For $B_{U+V}$ u have to ...
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23 views

Prove that then the linear mapping is surjective [on hold]

Let in the field $F$: $2 \ne 0$. Let $Q$ is a nondegenerate quadratic form on a finite-dimensional vector space $V$. Suppose that $Q(v) = 0$ for some nonzero vector $v \in V$. Prove that then the ...
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When the linear operator is self-adjoint?

Subspaces $V_1, V_2$ of the Euclidean space $E$ such that $E = V_1 \oplus V_2$. Under what conditions on $V_1, V_2$ the projection operator on $V_1$ parallel to $V_2$ is self-adjoint?
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Basis of $\mathbb{R}^{n}$. Nikolay Gusev's course on Stepic.

The exercise I'm given is: Consider the space $\mathbb{R}^{n}, n\geqslant3$. Let's take the column $(110...000)^{T}$ and by shifting all elements down by one cell we have $(011...000)^{T}, ...
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32 views

Condition for a matrix to have anticommutator

For a $N\times N$ complex matrix $M$, what's the necessary condition for the existence of a non-zero anti-commutator $N$, such that $\{M,N\}=MN+NM=0$ $?$ Is there a necessary and sufficient ...
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1answer
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the physical significance of the Lie Algebra of SE(3)

as we all know, the Lie group of $SE(3)$ can be written in the form of $4\times4$ matrix, say $$ \begin{pmatrix} R & t\\ 0 & 1 \end{pmatrix},\tag{1} $$ and its Lie Algebra, denoted as $se(3)$, ...
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Prove that $ 0 \leq B^T(C+B\mathcal{I}_V^{-1}B^T)^{-1}B \leq \mathcal{I}_V $

In Walter Zulehner's article Nonstandard norms and robust estimates for saddle point problems page 546 at the very top, he writes that $$ 0 \leq B^T(C+B\mathcal{I}_V^{-1}B^T)^{-1}B \leq \mathcal{I}_V ...
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1answer
50 views

Dimensions of a sphere and a ball

The volume of the unit ball in $\mathbb{R}^n$ is denoted by $v_{n}$ and the surface area of the unit sphere $S^{n-1}$ is denoted by $\omega_{n-1}$. What is the importance of writing $n-1$ and $n$?
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Finding the basis given a system of equations in R4

The question I'm given is this: Let $S$ be the subspace of $R^{4}$ consisting of the solutions to the following system of equations: $$x_{1}+2x_{2}+2x_{3}+2x_{4}=0$$ ...
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28 views

Question regarding systems of equations

If I have the following system of equations: $2+x^2-y^2=0$ $x^2-y^2-2=0$ And if I substitute $y$ by a function of $x$ and vice versa I get: $2+x^2-x^2+2=0$ $y^2-y^2-4=0$ I therefore get: ...
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1answer
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linear algebra - Matrix Algebra [on hold]

Let $C$ be $n \times n$ real matrix. Let W be the vector space spanned by $$I, C,C^2,C^3, \ldots,C^{(2n)}$$ The dimension of the vector space $W$ is (a) $2n$ (b) atmost n (c) $n^2$ (d) atmost ...
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solution of system of differential equation

Consider the system of ODE in $R^2$ , $Y'$ = AY , ${\bf Y(0)} = \pmatrix{0\cr 1\cr}$ , t>0 where A = $\pmatrix{ -1 & 1\cr 0 & -1 \cr}$ ${\bf Y(t)}$ = $\pmatrix{y_1(t)\cr y_2(t)\cr}$ ...
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1answer
17 views

Can matrices with dependent columns being QR factorization?

The problem comes from the $18.06$ Linear Algebra by MIT Open Courseware. The answer: I am very confused. According to the definition, Matrix A -> QR means that A has independent columns. BUT ...
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1answer
37 views

Why does a linear equation define a point-set of dimension one less than the space?

For example: If we are in 2-space (2 unknowns), a linear equation defines a line. If we are in 3-space (3 unknowns), a linear equation defines a plane. I mean, it seems obvious, but an explanation ...
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Sum of principal minors of order $n-1$ equals the sum of products of eigen values taken $n-1$ at a time

Let $A$ be a symmetric matrix of order $n$ .Prove that Sum of principal minors of $A$ of order $n-1$ equals the sum of products of eigen values taken $n-1$ at a time . Now if I consider the ...
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Projection from high dimension to lower, for visualization

I want to project high dimensional data points onto 2D screen coordinates, for visualization purposes. I want to be able to control the angles of projection manually (eg, with the mouse). I have ...
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Is a diagonalization of a matrix unique?

I was solving problems of diagonalization of matrices and I wanted to know if a diagonalization of a matrix is always unique? but there's nothing about it in the books nor the net. I was trying to ...
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Dual Norm proof

Let $\|.\|$ denote any norm on $C^m$. The corresponding dual norm $\|.\|'$ is defined by the formula $\|x\|' = sup_{\|y\|=1}|y^*x|$. (a)Prove that $\|.\|'$ is a norm? (b) Let $x, y \in C^m $ with ...
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MIT ocw math/computer science courses for a grade 11 student? [on hold]

I am 16 years old and I have decided to take the MIT math and maybe computer science courses online. I love math and computer science and I want to finish the learn the undergraduate courses as soon ...
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isolating x with two variables and negative exponents

I have: $$ 4^y = x^{-2} $$ Can someone hint to me what I need to do to isolate $x$? I'm not sure what to do.
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If the determinant of a matrix goes to infinity, does it means it has no inverse?

Context I have a linear time-invariant (single-input, single-output) system in state space representation (https://en.wikipedia.org/wiki/State-space_representation#Linear_systems): $$ \mathbf{x'}(t) ...
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Vectors sometimes used in math just as arrays/lists of numbers, sometimes as concept of “change”

As a freshman in a small town college. Ive been getting mixed signals to what vectors (and matrices/tensors) are. Sometimes I get the feeling they are used just as containers/arrays for multiple ...
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Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in ...
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Linear Algebra and Biology

Would it make more sense for a biology student to study two semesters of calculus during first year (dealing with derivatives in the first semester and integrals in the second) or to study one unit of ...
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Linear algebra: Solving for the coefficients on vectors

I am solving the following system: $$ -\frac{1}{r^2}\begin{bmatrix}\sqrt{\mu}\cos(\theta)\\ \sin(\theta) \end{bmatrix}= ...
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25 views

Bounding the off-diagonal entries of a matrix

The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & -i \\ i & 0 ...
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Show that all multilinear functions are proportional

This was the first problem on the final exam for my undergraduate Linear Algebra class almost 45 years ago. It still haunts me to this day. I think I lost my A to that question! I was caught off guard ...
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for every linear map $\ T:V\to V$ : $\ [T^*]_B=(M^t)^{-1}A^tM\ $ when $\ [T]_B=A$.

Let $V$ be an inner product space of finite dimension over $\mathbb{R}$ and Let $B=\{v_1,...v_n\}$ be a basis of V (not necessarily orthonormal). Let $M\in M_n(\mathbb{R})$ a matrix whose i,j ...
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Nilpotent Lie Algebras and 2-dimensional Lie Subalgebras

Let be $\mathcal{L}$ a finite-dimensional Lie algebra. How I can prove that if every $2-$dimensional Lie subalgebra of $\mathcal{L}$ is abelian, then $\mathcal{L}$ is nilpotent?
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Question in Introductory Linear Algebra [on hold]

I really need help with this question. I am in an introductory linear algebra course. If you guys could help me, I would really appreciate it. Here is the question: A large apartment building is ...
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How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$?

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$? I tried with $\det(A - aI) = (\cos\phi - a)^2 + \sin^2 \phi = 0$ and I got somehow to $2\cos\phi = a$, and I believe ...
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Tricks for quickly reading off the eigenvalues of a matrix

I noticed that some mathematicians have an uncanny ability to identify the eigenvalues of matrices without doing much in the way of computation. For instance, one might notice that all the rows have ...
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1answer
25 views

Given $v \in C^n$ that $u^Hu = 1$, and $D = iuu^H$ find all eigenvalues of $D$

Given $v \in C^n$ that $u^Hu = 1$, and $D = iuu^H$ find all eigenvalues of $D$ Well, I believe that $D$ is composed of orthonormal vectors, because of $u^Hu = 1$. Which means I believe that all ...
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What is the point of solving a system of linear equations using back-substitution (as opposed to reduced echelon form)

In lecture the other day, my professor offhandedly mentioned the existence of a process called back-substitution a way in which a computer program would solve a system of linear equations rather ...
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Find the spectrum of graph

Find the spectrum for the following graph by calcuation The spectrum of a graph $G$ is a list of the eigenvalues and the multiplicities of the eigenvalues of the adjacency of matrix $A$ of $G$. ...
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23 views

vector space homomorphism for $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$

I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields... It goes "Define a $\mathbb{F}_{5}$-vector space ...
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Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was ...
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Basis in Linear Algebra [on hold]

I am taking an introductory linear algebra course, and I am stuck on this problem: Explain why the set $W= \{(a,b,c)\ |\ a+b+c=0\}$ is a subspace of $\mathbb R^3$. After, find a basis for the ...