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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finding a basis for $W^\perp$ and understanding it.

Given $$ w_1 = \begin{bmatrix} 1 \\ -1 \\ 1 \\ 1 \end{bmatrix},w_2= \begin{bmatrix} 0 \\ 1\\2\\3 \end{bmatrix} $$ let $W$ be the subspace spanned by the given vectors. Find a basis for $W^\perp$ ...
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Fast way to switch between lagrange and newtonian representation of polynomials?

I just wanted to know whether there is a fast algorithm to switch between these two representation methods of polynomials $\in \mathbb{R}[x]$? By Lagrange I am refering to: enter link description ...
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97 views

Cauchy–Schwarz inequality properties

Let $V$ an inner product space Let $ F = C $ Let $u,v \in V$ I have to show that: if $\|v+u\| =\|u\| + \|v\| $ then $\exists c \in R$ $ c>0 $ such that $u=cv$ or $u=0$ or $v=0$ How come ...
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101 views

Vector spaces and finite dimensions related problem.

Please can you help me whit this problem. For $1.$ I did it as it's classical. What I am having trouble with are the other questions. Hints would be good but if you can explain that would be great. ...
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88 views

Proving that Hermitian conjugate operator is unique

A very basic question: Considering the $\Phi : Hom(U) \rightarrow M_{n}$ isomorphism between linear operators on vector spaces, and square matrices and the fact that Hermitian conjugate of an ...
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131 views

prove that $\operatorname{Ker}T=\operatorname{Im}(S)^\perp$ for some transformation

Question Let $V$ be an inner product space of finite dimension. Given linear transformations $T,S\colon V \to V$ such that $\langle T(v),w\rangle=\langle v,S(w)\rangle$, for all $v,w \in V$. Show that ...
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61 views

Finding an inner product

Question: Given 2 vector spaces $U=sp(1,1), W=sp(2,0)$. How do I find an inner product in $\Bbb R^2$ s.t. $U=W^{+}$ (orthogonal) I would love an explanation for the algorithm really, more than this ...
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1answer
60 views

Interpolating a linear transformation

I'm experimenting with some rudimentary ideas for data encryption (I've never formally taken a cryptology class). An idea that I had for an encryption was to use matrices. So I treat a data set as ...
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202 views

Using simple linear algebra for encryption?

e.g. the character $a = 97$ (it's computer decimal format, commonly known) and then using a pattern/key like $y = 31 x + 5$ to get $3012$ (substitute $97$ into $x, y$ is now the encrypted code). ...
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79 views

Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
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246 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
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81 views

Convert fundamental matrix Differential equations

I was wondering about the following: Assuming that I have found a fundamental matrix for a given ODE. How do I manage it, that the columns of the matrix $y_i$ fulfill the initial condition ...
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1answer
93 views

Cauchy–Schwarz inequality

So I have a doubt regarding of the way I proved something and I am not sure it is good. Let $V$ be an inner product space over $\Bbb C$ (the complex field). Let $y \in V$ and let $x = \lambda y$, ...
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1answer
196 views

verifying differential equation solution with sage

I solved the linear ODE system of equations: \begin{equation} x' = \begin{pmatrix}3&0&4\\0&2&0\\0&0&-3\end{pmatrix}x \end{equation} Skipping the details I got the following ...
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1answer
80 views

Algebra difficulties within an inverse stereographic projection problem

My book has the following question: Inverse Stereographic projection. Solve the equation $$x+iy = \frac{a+ib}{1-c}$$ for a, b, and c in terms of x and y. I found a really helpful book that works ...
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43 views

Simple question about equivalence of two forms of PCA as trace maximization over an implicit distribution

This may be a soft question of sorts. One formulation of principal component analysis is trace maximization: $$\arg\max_U \mathbb{E}_x \ [tr(U^Txx^TU)],$$ for $U^TU\le I$ and we assume that there is ...
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2answers
339 views

Estimating the transition matrix given the stationary distribution

Let's say we are given a Markov chain for variable $X = [x_1, ..., x_n]$; also we are given a desired stationary distribution for this graph $P_\infty = [p_1, ..., p_n]^\top$. How can we design an ...
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1answer
420 views

Finding the inverse of a matrix using elementary matricies

Can somebody help me understand what exactly is being asked here? I understand how to construct elementary matrices from these row operations, but I'm unsure what the end goal is. Am I to assume that ...
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117 views

Is the eigenvalue of this matrix really non-real ? If so How can I calculate this by hand?

I'm trying to find the Eigenvalues of the following $2\times 2$ matrix : $$ \begin{bmatrix}-2 & -7\\ 1 & 2\end{bmatrix} $$ I've been getting mixed results: by hand I've tried calculating the ...
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1answer
58 views

Computation of Eigenvalues

I am studying a linear algebra course and there's a problem with the calculation of the eigenvalues of a matrix. It's probably due to my own error, due to a wrong method or unsound algebraic concepts. ...
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3answers
191 views

What “is” a matrix in the context of a vector space?

I'm familiar with the definition of a vector space $V$ over a field $F$ I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space ...
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93 views

Show that the matrix $A+E$ is invertible.

Let $A$ be an invertible matrix, and let $E$ be an upper triangular matrix with zeros on the diagonal. Assume that $AE=EA$. Show that the matrix $A+E$ is invertible. WLOG, we can assume $E$ is Jordan ...
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137 views

Bounding the size of the product of a collection of sets

Reading a proof from some book I came across to the following argument that I somehow don't understand fully. A clarification would be very nice. Consider a collection $\mathcal{A},\mathcal{B}$ of ...
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74 views

Determinant characterization of subspace

Let $V\subset \mathbb{R}^n$ be a linear vectorial subspace of the euclidean space and $\{v_1,...,v_k\}$ a basis of $V$. I'm asking an easy characterization of the elements of $V$ using determinants ...
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74 views

Question regarding positive definiteness decompositions

The definition of positive definteness that I'm working with is: "A real n by n matrix $\mathbf{A}$ is positive definite if for all conformable, non-zero vectors $\mathbf{x}$ (n by 1), the following ...
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Is it always true that $\det(A^2+B^2)\geq0$?

Let $A$ and $B$ be real square matrices of the same size. Is it true that $$\det(A^2+B^2)\geq0\,?$$ If $AB=BA$ then the answer is positive: ...
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75 views

Quadratic Bezier curves representation as implicit quadratic equation

A quadratic bezier curve from points P1=(x1, y1) to P3=(x3, y3) with control point ...
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1answer
79 views

Polar decompostion should be a diffeomorphism, right?

I seem to have gotten stuck in the mud verifying what I thought was going to be a completely straightforward fact. I would appreciate if somebody could help dig me out. Inside the $n \times n$ ...
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1answer
48 views

Augmented matrices and system of equations

Why are $$[\mathbf{c}_1 \;\mathbf{c}_2]\begin{bmatrix} x_1\\ x_2 \end{bmatrix} = \mathbf{b}_1$$ and $$[\mathbf{c}_1 \;\mathbf{c}_2]\begin{bmatrix} y_1\\ y_2 \end{bmatrix} = \mathbf{b}_2$$ ...
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2answers
52 views

Change of basis and identity

Let $\beta = \{b_1,\dots, b_n \}$ be a base for $V$. Explain why the $\beta$ coordinate vectors of $b_1,\dots, b_n$ are the columns $e_1, \dots, e_n$ of the $n$ by $n$ identity. The solution ...
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51 views

how to generate parametrically a matrix of positive determinant

This is not about generating a positive-definite matrix, in which necessarily all eigenvalues are positive. I am interested in an algorithm that can generate any matrix with real entries and positive ...
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41 views

Why is $E_{\lambda}$ the kernel of the linear map $\alpha-\lambda I$

The book starts the chapter on Eigenvalues and Eigenvectors, and goes that this statement is obvious. Here $E_{\lambda}$ stands for the set of vectors $v$ such that $α(v) = λv$, for any scalar ...
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338 views

Proving a set of linear functionals is a basis for a dual space

I've seen some similar problems on the stackexchange and I want to be sure I am at least approaching this in a way that is sensible. The problem as stated: Let $V= \Bbb R^3$ and define $f_1, f_2, ...
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2answers
172 views

Is determinant uniformly continuous?

The determinant map $\det$ sending an $n\times n$ real matrix to its determiant is continuous since it's a polynomial in the coefficients. Is it also uniformly continuous?
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172 views

What does is mean by differentiate a matrix $E - DB^{-1}C$?

The problem I am trying to solve is: Prove that the set of $m \times n$ is matrices of rank $r$ is a submanifold of $\mathbb{R}^{mn}$ of of codimension $(m - r)(n -r)$. [HINT: Suppose, for ...
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434 views

Are two matrices of the same rank similar?

I know that if two matrices $A$ and $B$ are similar implies that they have the same rank. However, if they have the same rank are they similar?
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142 views

Upper triangular matrices proof

An excerpt of the proof from Axler I am trying to understand a simple statement that $\lambda_1 = 0 \implies Tv_1 = 0$ If $T = \begin{bmatrix} \lambda_1 &1 \\ 0 & \lambda_2 ...
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1answer
137 views

Necessary and sufficient condition for Nilpotent Matrix

Let $\alpha$ and $\beta$ be nonzero elements in $\mathbb{F}(n,1)$. Then $A = \alpha\beta^T \in \mathbb{F}(n,n)$ Prove that a necessary and sufficient condition for $A$ to be nilpotent is ...
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derivative of matrix [duplicate]

Is the nonsingular matrices open? How can I show that every $m \times n$ matrix is in the image of the derivative of an $m \times n$ matrix (how to differentiate it?) Thanks
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81 views

Factorizing a matrix into a matrix and its transpose

Let $W\in \mathbb{R}^{n \times n}$ be a positive semi-definite matrix. Then, what are some well-known factorization methods that guarantee $W=A^T A$, with the conditions being that \begin{align} ...
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77 views

Applications of companion matrices?

I'm looking for interesting applications of companion matrices, I can also use the Frobenius Normal Form. I already covered the Cayley-Hamilton Theorem and the application to linearly recursive ...
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2answers
597 views

diagonalize quadratic form

I have this quadratic form $Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$ And they ask me: for which values of $x,y$ and $z$ is $Q=0$? and I have to diagonalize also the quadratic form. I calculated ...
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110 views

Can we conclude from $V=\ker(T) \oplus\operatorname{im}(T)$ the invariance of both subspaces?

Can we conclude for an endomorphism $V \in \operatorname{End}(V)$ where V is a finite dimensional vector space from $V=\ker(T) \oplus \operatorname{im}(T)$ that nullspace and image are invariant ...
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1answer
40 views

Forming the $N(A)$

Suppose we have the following solutions $\begin{bmatrix}-3\\ 1\\ 0\\ 1\end{bmatrix},\begin{bmatrix}-2\\ 0\\ -6 \\1\end{bmatrix}$ to $Ax=0$ (nullspace) that makes a 3 by 4 matrix true. How would you ...
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160 views

The tangent plane of orthogonal group at identity.

Why the tangent plane of orthogonal group at identity is the kernel of $dF_I$, the derivative of $F$ at identity, where $F(A) = AA^T$? Thank you ~
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260 views

Prove that if $\langle x,z\rangle = 0$ for all $z$ then $x=0$

Hi I just wanted to check if my reasoning in this proof was correct. THe question is: let $\beta$ be a basis for a finite dimensional inner product space a) prove that if $<x,z> = 0 $ ...
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235 views

Proving that a Particular Set Is a Vector Space

Let $V$ be the set of all differentiable real-valued functions defined on $\mathbb R$. Show that $V$ is a vector space under addition and scalar multiplication, defined by $$(f+g)(t) = f(t) + ...
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71 views

change of bases - matrix representation of linear maps

I am trying to solve a problem and got stuck. I suppose I made a stupid mistake somewhere, could somebody explain where? Let $B = \{v_1, v_2, v_3\}$ be a basis of a vector space $V$, and let $B' ...
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2answers
84 views

Equivalent definitions of Verma modules

This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let $ \lambda $ be a weight of $ ...
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3answers
169 views

If the Wronskian of two functions is zero then these functions are LD

I'm studying a book of differential equations which says that if the Wronskian of two functions is zero then these functions are linearly dependent. the author doesn't prove it, he simply said as a ...