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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Any way to simplify the matrix integral $\int_{0}^{\infty}e^{A z}e^{B z}dz$ if A and B do not commute but are diagonalizable>

Define $A$ and $B$ square matrices where all eigenvalues are $< 0$ for both, and there is no eigenvalue multiplicity. Completely diagonalizable, etc. But assume that $A$ and $B$ do not commute. ...
2
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1answer
82 views

How does nested notation work?

Here is dot notation: $$ \overline a . \overline b = (a _x * b _x) + (a _y * b _y) $$ $$ (a _x, a _y) . (b _x, b _y) = (a _x, b _x) + (a _x, b _x) $$ Here is another way to describe it with sigma ...
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124 views

Find the Jordan form of the matrix

let $A\in F^{8x8}$ , $A^3=0$,$A^2\neq0$ Find its Jordan forms that possible. solution was : characteristic polynomial $P_A(x)=x^8$and ...
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1answer
42 views

Solution to this system of equations

I'm currently reading the Michael Artin book "Algebra" and at page 4 I've encountered a detail, that's not totally clear. Here's the relevant excerpt from the book. Thus the matrix equation ...
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1answer
58 views

Equation to describe dot notation

Only learned what sigma notation actually was yesterday so bear with me! Basically, dot notation can be described like this: $$ \overline a . \overline b = (a _x * b _x) + (a _y * b _y) $$ So I was ...
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1answer
73 views

Elementary proof that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$ [duplicate]

Prove that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$ Since $Gl_n(\mathbb R)$ is homeomorphic to an open subset of $\mathbb R^{n^2}$, this boils down to proving that two open ...
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1answer
318 views

Norm of Block Diagonal Matrix

Given a matrix $A \in R^{m \times n}$ with known upper bound on the operatornorm $\| A \|$ I want to find an upper bound for the operator norm of the square root of the following matrix that is given ...
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65 views

Principle of superposition for linear systems

I have always learned about the superposition principle for systems of the form $$\dot x = A x + B u,$$ $$ y = Cx.$$ where $A$, $B$, and $C$ are time invariant matrices of appropriate dimensions. ...
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1answer
29 views

Show $\textrm{Tr}(f\circ f^*)\ge 0$ for euclidian/hermitian space $(V,<,>)$

How can I show that for an euclidian/hermitian space $(V,<,>)$, for every endomorphism $f:V\to V$ and the adjoint map $f^*$ the inequality $\textrm{Tr}(f\circ f^*)\ge 0$ is valid and only when ...
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29 views

How to test if $m$ vectors are linearly dependent when they are $n$ dimensional and $m < n$

I'll be shocked if this isn't a duplicate, but I haven't had a lot of luck finding an answer to this so far. How do you test if a set of vectors $v_1, \ldots v_m \in \mathbb{R}^n$ are linearly ...
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16 views

Question about bounding PSD matrices.

Given a positive definite matrix $A$ and a non-square, full rank matrix $B$ such that $B^T B$ is positive definite. Is it true that: $$ trace \big((B^T A B )^{-1}\big) \ge \frac{1}{\lambda_{max}\{B^T ...
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1answer
151 views

Positive definiteness of power of distance matrix

I have a matrix whose entries are powers of Hamming distances between binary vectors: $a_{ij} = s^{ d(x_i,x_j) }$ where $0\le s\le 1$ and $x_i$ an $n$-dimensional binary vector and $d$ is the ...
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1answer
69 views

Is there a symbol for the number of dimensions in a vector?

Here's an equation from a text book for computing a unit vector: $$\hat v = \frac{\overline v}{ \sqrt{ \sum ^n _{i=1} (\overline v_i)^2 } }$$ Now I may be wrong here, but using $n$ doesn't really ...
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1answer
111 views

Indecomposable quiver representations

Is there are any way to found indecomposable representation of a given quiver explicitely if it's dimention vector is given?
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67 views

Conditions for a positve vector in null space

Precursor to problem: Preparing for an exam in General Equilibrium Theory. One of the important theorems is the No Arbitrage Theorem which states that for an economy with $S$ states of the world and ...
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1answer
64 views

Linear transformations, $R^5,R^4$

Is there a Linear Transformation $T : R^5 \rightarrow R^4 $ that its $KerT$ is $KerT = \{( x,y,z,t,w) \in R^5 | x = 2y ,and, z = 2t = 3w\}$ Well, I tried to prove that by first saying : $KerT = ...
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36 views

Is there any significance to this matrix/operator?

I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on $\mathbb{R}^2$ $[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 ...
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2answers
43 views

Similar Matrices and Linear Transformations

There is a theorem in the course notes but the lecturer didn't given the proof: If $A,B\in M_n(K)$, where $K$ is a field. Then $A\sim B$ if and only if they represent the same linear transformation ...
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1answer
39 views

Problem understanding dual optimization problem?

I am reading this paper: http://dl.acm.org/citation.cfm?id=1390696 Following optimization problem is defined in section 2: \begin{align} \max_{\mathbf{X}>0} \log ...
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31 views

Matrix similarity & Characteristic polynomial (linear algebra)

I need to practice this kind of material, so as a self practice, I though about this: A1 and A2 are both nxn matrices. A1 is Invertible matrix. A1A2 & A2A1 are necessarily similar? and how about ...
2
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1answer
68 views

Why do special solutions of $Ax=0$ form a basis for null-space of $A$?

I read somewhere that The $n-r$ special solutions of a $m \times n$ matrix with rank $r$ form a basis for its null-space. If we consider the general RREF for the given matrix, it has the form: ...
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1answer
150 views

I am having a algebra exam soon I found this on an exam paper and have no Idea how to do it. Can anyone help

We say that a matrix $\mathbf{Q}$ is orthogonal if $\mathbf{Q}^T = \mathbf{Q}^{−1}$, recall $\mathbf{Q}^{T}$ is the transpose of $\mathbf{Q}$. Show that the dot product of any two vectors ...
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91 views

Characteristic polynomial and matrix over finite field

Consider the finite field $\mathbb{F}_2$ and all polynomial of degree $n$ over the finite field. So there are $2^{n+1}-1$ number of possible non-zero polynomials. It is needless to mention that there ...
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3answers
66 views

Linear Algebra - Diagonalizable matrix

It's a new topic we learn during the linear algebra class and I need a bit help understanding. Lets say, for example, that I have this matrix: \begin{pmatrix}2&1\\x&8\end{pmatrix} and x ∈ R ...
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1answer
509 views

Find the distance between two lines given in parametric form (should be easy).

I am struggling with this problem in my math book, I cannot seem to get the calculation right even after MANY attempts. I have solved the question with other solutions but this one should also work. ...
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1answer
53 views

linear space proof

The question is that V is the span of these vectors in the diagram b2,b3,b4. Please help me in this problem, I know all the theory that for it to be a linear space it should be closed under addition ...
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2answers
32 views

Evaluating determinant of an implicit matrix

I know that row operations does not change the determinant of a matrix but I also know that for example, A is a nxn matrix and if det(A) = 2 then, det(2A) = (2^n)*det(A). So, how should I approach ...
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2answers
52 views

What does this notation mean? Eigenvectors and Eigenvalues

I am trying to solve this question but I couldn't understand what does $A(v_1+v_2)$ mean. Could anybody help me with this? Thanks...
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2answers
80 views

Relation between basis of a vector space and those of kernel, image of a linear transformation

Let $T:V\rightarrow W$ be a linear transformation. If we have a basis of V such that a part of basis mapped under T gives a basis of the image of T, is it true that the rest of the vectors in the ...
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40 views

Definition of linear independence when $v_i=0$

I have read that linear independence occurs when: $$\sum_{i=1}^n a_i v_i =0$$ Has only $a_i=0$ as a solution, but what if all $v_i$ were $0$ then $a_i$ could vary and still yield $0$. Does that mean ...
2
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1answer
178 views

Regular pentagon vector proof

Given that $v = DC = \lambda EB$, prove that $\lambda v = CB + ED$. Whatever I try seems to end up with $CB + ED = (\frac {1}{\lambda} - 1)v$, ie: $$CB + ED = CD + DE + EB + ED = EB - DC = EB - ...
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1answer
149 views

finding the Rank and basis of null space of this matrix

Please help me with this question. The question is to find the rank of the matrix and then the basis of the null space, I first put the matrix A in reduced row echelon form and then I wrote the ...
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1answer
93 views

How to prove $ |\langle u,v\rangle| \leq ||u||||v||$

How to prove $ |\langle u,v\rangle | \leq ||u||||v||$ Note: I have given this many attempts so don't downvote due to lack of effort, refer to edit history for evidence of said effort
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1answer
48 views

What is the distribution of $x^TAx$ when $x$ is gaussian ($A$ may be not symmetric)

Suppose that $A \in \Re^{d \times d}$, $x \in \Re^d$ and each component of $x$ is independently sampled from $N(0,1)$. I wonder what is the distribution of $x^TAx$. To be more concrete, how will the ...
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288 views

Subspace of a finite dimensional inner product space, independence of basis choice

Let $W$ denote a subspace of a finite dimensional inner product space $V$, and let $$\beta = \{w_1,w_2,\dots,w_r\}$$ denote an orthogonal basis for $W$. For any $v\in V$ define $$proj_{\beta}v = ...
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1answer
203 views

Given a skewed ellipse, how to determine its axis lengths?

I am mentoring a student who is working on a library to import Adobe Pagemaker documents into LibreOffice. Pagemaker represents ellipses as a bounding box (of the original, untransformed ellipse) and ...
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1answer
71 views

Coordinates of octahedron's vertices and checking if a point is inside it.

Given that I have the distance between the center of an octahedron and any of its faces (regular octahedron, so all the distances are equal), how can I calculate the coordinates of its vertices, ...
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2answers
107 views

Linear Space in Vector Spaces question.

How do I do the first part of the question where they say V1 U V2 is not a linear space, please help my exam is very close. In the marking scheme it says it's not closed under addition but can ...
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0answers
85 views

Prove solution does not exist for inequalities system

I have an inequalities sytem like the following: Example > x+y+z <= A > x+y <= B > x+z > C > y+z > D > x >= E Let A,B,C,D,E be any ...
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248 views

Convergence of Matrix Power Series

If $A$ is a square matrix with complex entries, then $\| A\|$ is defined as the sum of the absolute values of the entries of $A$. I have shown that this matrix norm is homogeneous, subadditive, and ...
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1answer
32 views

Finding linear transform matrix from characteristic polynomial

I got two similiar very simple question on a notebook. 1)let characteristic polynomial $P_A(x)=x^2+2x-3$ and $T:V\to V$ and DimV=2,S={$\alpha_1,\alpha_2$} is ...
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1answer
73 views

How to determine the visibility of an object from the top of a hill

We are developing software to train children how to cross the street safely. Part of the training is to teach them not to cross when they don't have enough visibility due to obstacles. In this case, ...
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1answer
56 views

Transpose of a differential operator

Let $H$ be a diagonalizable matrix (not necessarily Hermitian). Then, it induces a biorthogonal left and right vectors, such that $$ ...
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174 views

example of non linear matrix transformation

Could you give me an example of non linear transformation matrix? What is the difference between linear and non linear transformation matrix?
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227 views

How to understsand eigenvalues and eigenvectors.

I know basic linear algebra (what is a matrix, what is a determinant, what is a square matrix, what is an inverse of a matrix, how to add/sub/multiple matrices etc.) But I am finding the concept of ...
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1answer
55 views

Projection onto Polyeder

I know how to projects onto a linear subspace of $\mathbb R^3$, but how to project a point $x$ onto an polyhedron given as the intersection of three halfspaces $$ \langle y_1, x \rangle \ge c_1 ...
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1answer
251 views

Can someone explain the effects of degenerate basic feasible solutions in the simplex algorithm?

I was given this on an assignment sheet, and am now using it to revise from...I cannot remember the issues that arise from degeneracy of basic feasible solutions... Let $P$ =$\{x\in \mathbb{R}^n ...
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45 views

Looking for ranks.

Let $X$ belong to $\operatorname{Mat}_n(R)$, where X is inrevertible and let its column vectors be $X_1,X_2, \dots X_n$. Let $Y$ be a matrix that has the column vectors $X_2, X_3, \dots , X_n$. Let ...
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95 views

Find an orthonormal basis of a particular bilinear form

Let $V=\mathbb{R^3}$. Find an orthonormal basis in which the bilinear form with matrix $A$: $\begin{pmatrix} 2 & -2 & 0 \\ -2 & 1 & -2 \\ 0 & -2 & 0\end{pmatrix}$ has a ...
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1answer
52 views

Solution of $B^{\text{T}}B=Q$ for B?

Let $B$ be an $m$ by $n$ matrix whose entries are either 1 or 0 (it is an undirected incidence matrix). Given the $n$ by $n$ matrix $Q$, defined by $B^{\text{T}}B=Q$, is it possible to solve for $B$? ...