# Tagged Questions

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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### If A is 5 by 3 and B is 3 by 5 (with dependent columns), Is $AB = I$ impossible?

Let me first introduce the problem. This is part of the quiz problem from MIT's 18.06 course (Spring 2012 semester, quiz 1, problem 3). My question is related to (b) but (a) is mentioned in the ...
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### If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$.. [on hold]

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$ dimensional subspace of $\mathbb{R}^n$ that's contained in $H$, show that $G = H$ I know that a subspace of $\mathbb{R}^n$ is ...
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### Calculation of $\frac{a_{20}}{a_{20}+b_{20}}$?

The solution of $\frac{a_{20}}{a_{20}+b_{20}}$ is $-39$ (This is wrote by answer sheet) from the recursive system of equations : \begin{cases} a_{n+1}=-2a_n-4b_n \\ b_{n+1}=4a_n+6b_n\\ a_0=1,b_0=0 \...
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### (Double) Coset of $GL(n, q^2)/GL(n, q)$

I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the ...
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### calculate centroid of triangle on a graph

Given ANY three points on a graph that form a triangle, how do you find the centroid using geometry? So basically I have three points (X1, Y1), (X2, Y2), and (X3, Y3). I am trying to use the slopes ...
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### Show that the trace of A is less than n

Let $A$ be an $n\times n$ matrix with complex entries such that $A^k=I_n$ for some positive integer $k$. Show that the trace of $A$ satisfies $$|tr(A)| \leq n.$$ I have no idea how to approach this ...
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### Theorem regarding direct sums

Let $w_1$ and $w_2$ be subspaces of V. Prove that V is direct sum of $w_1$ an $w_2$ iff each vector in V can be uniquely written as $x_1 + x_2$ where $x_1$ belongs to $w_1$ and $x_2$ belongs to $w_2$ ...
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### Conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z}) × GL_m(\mathbb{Z}/q\mathbb{Z})$ [on hold]

Find the number of conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z})× GL_m(\mathbb{Z}/q\mathbb{Z})$ of cyclic subgroups of order pq?.
I ran into this question when writing a program. I need to generate two matrices, and calculate their product. However, I must ensure all entries are within 8-bit signed integer range, i.e. $[-128, ... 1answer 547 views ### MATLAB determining elementary matrices for LU decomposition I am confused by this question I am studying for MATLAB practice. 0answers 16 views ### Water drop evaporation time and contact angle I'm measuring water drop evaporation on different surfaces and it would be nice to have an equation to roughly estimate evaporation time (or contact angle). Some drops are hydrophobic, others ... 0answers 35 views ### What is a linear isomorphism? I am working with the book Manifolds and Differential Geometry from Lee and I am a little bit puzzled since he sometimes talks about linear isomorphism (proposition 2.3 for example). But isn't an ... 1answer 55 views ### Prove that the ellipsoid$x^T W x \leq 1$is invariant under$f (x) = A x$[closed] Given matrix$W \succ 0 $and a set$\mathcal{Z} := \{z \mid z^T W z \leq 1\}$, prove that if$Az \in \mathcal{Z}$and$z \in\mathcal{Z}, then the following inequality holds A^T W A - W \... 1answer 379 views ### Linear system with positive semidefinite matrix I have a linear system Ax=b, where A is symmetric, positive semidefinite, and positive. A is a variance-covariance matrix. vector b has elements b_1>0 and the rest b_i<0, for all ... 1answer 75 views ### Is it true that SL(n, \mathbb R)=<\{ABA^{-1}B^{-1} : A,B \in GL(n,\mathbb R) \} >? [closed] Is it true that \operatorname{SL}(n, \mathbb R)=\left\langle \left\{ABA^{-1}B^{-1} : A,B \in \operatorname{GL}(n,\mathbb R) \right\} \right\rangle? 1answer 26 views ### Basic questions about optimizing concave function with constraints Consider the following problem: \begin{align} {\tt Maximize} \quad M(\mathbf y)& = \log \Big(\prod_i U_i(y_i) \Big) \\ y_i & = \sum_{j=1}^{m} \frac{x_{ij}}{a_{ij}} \\ \sum_{i=1}^{n} \frac{c_i ... 0answers 12 views ### Curious about the deduction procedure on shannon transform and stieltjes transform I found some scholars clarified that the relationship between the Shannon transform and Stieltjes transform with \frac{\gamma}{\log e}\frac{d}{d\gamma}\mathcal{V}_N(\gamma) = 1 - \... 1answer 31 views ### Least Squares Algorithm with Inverse Norm Given an overdetermined linear system A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{m \times 1} with A < 0 and b < 0. What is a good way to numerically determine \min_x \left\lVert ... 0answers 39 views ### what is Expected Mean Thus the expected mean\mu$of the set$\mathcal Scan be given as \begin{align*} \mathbb E \mu&= \sigma^2+\frac 1r \sum_{i=1}^m\left(\mathbb E\lambda_i-\sigma^2\right)\\ &\geq \sigma^2+\... 1answer 32 views ### Proof Determinant of Block Matrix does not depend of a variable I have the following matrix (called BRM): BRM = \begin{bmatrix} -A & A & \mathbb{0}_{3\times3} & B & \mathbb{0}_{3\times1} & \mathbb{0}_{3\times1} \\ -C &...
Given a matrix $A\in \mathbb{C}^{n\times m}$, clearly we can write $A=\Re(A)+i \Im(A)$, i.e., the real and imaginary part of $A$. (For instance, $A=[1,i]$, then $A=[1,0]+i[0,1]$). I am interested in ...