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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Relative Eigenvalue Perturbation Bound deduction from Ostrowski's Theorem

I need to deduce the relative eigenvalue perturbation bound from Ostrowski's Theorem. In short i need to proove ´this statement; $\frac{|\lambda_k(SAS^*)-\lambda_k(A)|}{|\lambda_k(A)|} \leq ...
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236 views

Every vector space has a basis using minimal spanning set.

We have seen the argument for proving the above statment using Zorn's Lemma by asserting the existence of a maximal linearly independent set which serves a basis. In finite dimensional vector space, a ...
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53 views

proof non diagonalizable matrix is not an inner product

Given $ A \in M_n(\Bbb C) $ and $ <x,y>_A = x^TA\overline y $ I need to proof that if A is non diagonalizable then $<.,.>_A$ is not an inner product. I thought about: Let A be non ...
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87 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
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37 views

Is this functional linear?

I know it's trivial, but is this functional not linear? $\phi:\mathbb{R}[X]\ni p \rightarrow p(0)p \in \mathbb{R}[X]$ $$\phi(p+q)=(p+q)(0)\cdot(p+q)=(p(0)+q(0))\cdot(p+q)\ne\phi(p)+\phi(q)$$
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72 views

Why must P be orthonormal, and not just orthogonal, for change of variable in Quadratic Form? [Kolman P560 8.8.24]

Lay P402 : A change of variable is an equation of the form $x=Py$, where $P$ is an invertible matrix and $y$ is the (neW) coordinate vector of $x$ relative to the basis of $\mathbb{R}^{n}$ determined ...
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49 views

Why Riemannian metrics defining the same angles are conformal [duplicate]

Suppose $g_1$ and $g_2$ are two metrics defining the same angles, which means $g_1(X,Y)/(g_1(X,X)g_1(Y,Y))^{0.5}=g_2(X,Y)/(g_2(X,X)g_2(Y,Y))^{0.5}$ for all pairs of vector $X,Y$.I want to prove that ...
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31 views

tensor product of infinite dimensional vector spaces [duplicate]

Let $V,W$ be vector space over field $k$. Then \begin{eqnarray*} V^*\otimes W &=&V^*(\oplus_{i\in I} k_i)\\ &=&\oplus_{i\in I}(V^*\otimes k_i)\\ &=& \oplus_{i\in I}V^*_i\\ ...
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81 views

Average Line of a Set of Lines

Suppose we have 10 lines in an x-y plane. The lines are somewhat clustered together, and going more or less in the same direction. The data I have for these lines is their line equation: $$y = a + ...
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Find values of $p$ for which in a field $\mathbb{Z}_p$, two equations have 1 solution.

Find values of $p$ for which in a field $\mathbb{Z}_p$, two equations, say $7x-y=1$ and $11x+7y=3$ have 1 solution. I can give some values of $p$ like the obvious $p = 7, 11$. But how do I ...
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52 views

Product of PD matrices is PD, once again

I see on wikipedia that the product of two symmetric positive definite matrices is also positive definite if the resulting product is normal. I've seen the proof for the same case with PSD matrices in ...
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183 views

Cauchy-Schwarz Inequality - Proof using Quadratic Polynomial [Lay P379 Thm 6.7.16]

I don't perceive https://www.dpmms.cam.ac.uk/~wtg10/csineq.html, about why it " is an obvious thing to write down" "a quadratic form, use the fact that it is non-negative everywhere, and ...
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78 views

Hermitian Matrix with their eigenvalues arranged in non-decreasing order

I need to formulate one property of Hermitian Matrices. It goes like this; If A, B $\in M_n$ are hermitian and their eigenvalues are arranged in non-decreasing order , then $\lambda_i(A+B)\leq ...
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53 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
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26 views

Proving $\dim(\ker( p (T) ) ) = n\cdot d$ where $n$ is a positive integer and $d$ is the degree of the polynomial.

A few more details: $T$ is $T : V -> V$ for some space $V$. Also, the polynomial $p$ is irreducible where $d \ge 1 $. What I've done so far was to restrict the transformation to the invariant ...
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4answers
172 views

Proof that $e^x$ is the eigenvector or the derivative operator

I remember hearing my professor talk about how $e^x$ shows up in all our differential equations because it is the eigenvector for the derivative operator. Can someone explain and prove this to me? I ...
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1answer
501 views

Finding Eigenvalue for cubic equation

I'm learning finding eigenvalues. I learned how to find simplistic eigenvalues for $3\times3$ matrix. By using below way. With this way I can only solve if I have simple determinant equation, like ...
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73 views

Transformation of 2D profile to 3D coordinates

I am sure that answer for similar questions have being given more than one thousandth times, but correct answer that suits my needs I still haven't found. Currently I am developing simple 3D app. My ...
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1answer
151 views

how to relate the eigenvalues and eigenvectors of these two matrices?

If $W, Y \in R^{n \times n}$, then how the eigenvectors and eigenvalues of these two matrices are related? $C = W +iY, B = \begin{bmatrix} W & -Y\\ Y & W\\ \end{bmatrix} $ Specifically, ...
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77 views

categorification and linear algebra

Vect is the category with objects as vectors and arrows as linear transformations between them. Then these arrows have quite a bit of structure. We can take the transpose, trace, determinant, ...
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49 views

A problem on unitary matrices

Let $A, B \in M_{n \times n} (\mathbb{R})$. Suppose there exists an unitary matrix $U \in M_{n \times n}(\mathbb{C})$ such that $$U^{-1}AU = B.$$ Show that there exists invertible symmetric $S \in ...
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61 views

Linear Algebra: Finding the basis of a subset

Consider $W=\{(v,w,x,y)\ |\ 0=2w-x+y+v\}$ $\subset$ $\mathbb{R}^4$ i) Show that $W$ is a subspace for $\mathbb{R}^4$ ii) If $W$ is a subspace, find the basis for $W$ I get how to do problems where ...
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27 views

Markov transition matrix for given problem

I am working on the following problem (in an introductory linear algebra course): Every decade 15% of people in rural areas move into urban areas, and 5% of urban dwellers move into rural areas. What ...
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41 views

Projection on cone of non-negative definite matrices

Ok, so if you have a real symmetric matrix $Q$ then the projection of that matrix on the cone of symmetric non-negative definite matrices $\mathcal{C}$ can be explicitly found if we do an ...
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1answer
64 views

Given rank($A$) = rank$(A^{2})$, can we prove A is invertible?

I am trying to prove, given rank($A$) = rank($A^{2}$), that Nul(A) = Nul($A^{2}$), and that the intersection of the column space of A and the null space of A is the set containing the zero vector. If ...
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5answers
185 views

Algebra problem stumping me

I have recently run into an algebra problem that goes as follows. Using the digits $1$ to $9$, $$ \left\{ \begin{align} A + B + C + D &= EF \\ E + F + G + H &= CJ \\ B + G + J ...
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24 views

Finding basis of $R(T)$

Let $T: \text{Mat}_{2 \times 3} \rightarrow \text{Mat}_{2 \times 2}$ be defined by $$T \left(\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}\right) ...
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109 views

Find the axis of rotation from the rotation matrix.

This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I ...
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132 views

Mapping unit sphere to ellipsoid

Consider an $N$-dimensional space. Let $M$ be a square $N\times N$ (real, but I am interested in complex case too) matrix. Are the following (hyper)ellipsoids (or degenerate hyperellipsoids)? $\{v ...
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220 views

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W This is something from a practice sheet I got. I'm studying for a linear algebra final. I am unsure if we have ...
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1answer
42 views

Given ρ and σ are commuting projections, show ker(ρσ) = ker(ρ) + ker (σ)

Here's the prompt: Given $\rho$ and $\sigma$ are commuting projections, prove that $\rho \sigma$ is a projection and show that $\operatorname{ker}(\rho \sigma) = \operatorname{ker}(\rho) + ...
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128 views

Matrixes and modulo of a vector

Consider an $N$-dimensional space. Consider the function $\kappa$ which maps a square $N\times N$ matrix $M$ into the scalar field $v\mapsto \lvert Mv \rvert$ (for $v$ being a vector). Is the ...
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44 views

Show two vectors are linearly independent

So I need help with this problem! I am confused because there is only one equation? I tried writing it in form $af(x) + bg(x) = 0$ but I really am quite stuck. Any help is greatly appreciate.
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292 views

Can Moore–Penrose pseudoinverse solve for underdetermined linear system?

Thanks for reading my thread. I am thinking, many of us know that Moore–Penrose pseudoinverse can solve for overdetermined system $Ax=b$, where $x=(A^TA)^{-1}A^Tb$; for exmplae the linear regression ...
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28 views

Simultaneous iteration of Symmetric Matrices

Given a Matrix $A$ we can use Simultaneous iteration(Using power iteration on all columns simultaneously) to compute the d biggest eigenvalues. Now this method will give you the biggest eigenvalues, ...
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1answer
32 views

What is meant by $\langle \cdot,\cdot \rangle ^H_\mathbb{R}$?

there is the following statement: Let $\langle \cdot,\cdot \rangle_\mathbb{R} = \sum_{k = 1}^{n} x_k y_k$ be that standard Euclidian scalar product in $\mathbb{R}^n$ and $\langle \cdot,\cdot ...
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54 views

Question on Hamming distance

Let V_n be n-dimentional vector space over GF(q). E is k-dimentional vector subspace which is a linear q-ary (n,m,d) code and also consider the radius e = [(d-1)/2]. Assume that E is not a perfect ...
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65 views

Find the basis of a transformation matrix for an endomorphism

I have a 3x3 transformation matrix $D_{BB} (f)$ with $B$ as a basis of vector space $V$ and $f$ as a diagonalizeable endomorphism $f : V \to V$ given. Basis $B$ is not explicitly given. The entries of ...
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138 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
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34 views

A question about linear independence

Let $S$ be a linearly independent set. Let $S'$ be a proper subset of $S$. $Span$$(S') \neq Span(S)$ Let $v$ be an element of $S$ which is not contained in $S'$; such an element must ...
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107 views

wedge product of $m$ vectors in $\mathbb{R}^n$

I came across the symbol $|v_1 \wedge \dots \wedge v_m|^{-1}$ in a paper - this is the norm of the wedge product of vectors $v_k \in \mathbb{R}^n$ . I thought it's meaning was self-evident until I ...
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Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
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Prove that the mapping $\psi : L(V,W) \rightarrow L(W^*, V^*)$ given by $\psi(T) = T^t$ is an isomorphism.

Let $V,W$ be finite-dimensional vector spaces over the same field $\mathbb{F}$ and let $L(V,W)$ be the vector space of $\mathbb{F}$-linear transformations from $V$ to $W$. Prove that the mapping ...
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1answer
48 views

plot phase portrait in matlab

I have solved the first part of this question. can any one please help to plot this using ODE in matlab. at least for one condition.
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34 views

concept between group and vector space, compare G/N with V/W

When we considered factor groups G/N, we need N to be normal,but in vector space V/W, why W only be subspace?
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29 views

Gaussian Elimination theoretical question

You know how Gaussian Elimination can be broken up into a sequence of L-U premultiplications right? Suppose that there is a matrix $A=a_{i,j} : j=1,...,n$ is an $n × n$ real matrix such that ...
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641 views

Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues - Proof Strategy [Lay P397 Thm 3]

Herein, I denote the Hermitian conjugate by * (ie: $A* = \bar{A}^T) $. Let $v_i$ and $v_j$ be two eigenvectors of an Hermitian matrix H. First of all suppose that their respective eigenvalues i and j ...
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43 views

A problem regarding geometric progressions

Hello my homework included this problem and I really need a hint how to solve it. It says that the numbers $a_1,a_2 \ldots a_n$ form a geometric progression. Knowing $S=a_1+a_2+\ldots+a_n$ and $P=a_1 ...
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183 views

Let f : X → Y and g : Y → Z be bijective mappings. Show that gf is bijective

Let f : X → Y and g : Y → Z be bijective mappings. Show that gf, the composition of f and g, is bijective. I have that since f(x)=y, and g(y)=z we get g(f(x))=g(y)=z is this enough to show gf is ...
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1answer
261 views

Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...