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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Proving that exists only one basis which is dual to a given basis

Question Let $V$ be a finite dimensional vector space over $\Bbb F$ and $V^*$ it's dual space. Let $f_1 ... f_n$ be a basis for $V^*$. Prove that $\exists ! e_1 ... e_n$ - basis for $V$ s.t. $f_1 ...
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3k views

Finding an orthonormal basis using Gram Schmidt process

OK, here's a question with polynomials. We want to find an orthonormal basis using Gram Schmift. Assuming that we are in a vector space V, $R^2[X]$ where {$f = \lambda_0+\lambda_1X+\lambda_2X^2$}. ...
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56 views

ODE solution with variation of constants verification

I am not sure how to verify my solution is correct as the matrix $A$ is a function of $t$. I want to solve the following IVP with the variation of constants formula. \begin{equation} ...
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3answers
162 views

What is wrong with this Jordan normal form computation?

The question I am working on is to compute the Jordan normal form of $$A := \begin{pmatrix} 2 & 1 & 5 \\ 0 & 1 & 3\\ 1 & 0 & 1\end{pmatrix}.$$ The characteristic polynomial and ...
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1answer
617 views

Understanding a Gram-Schmidt example

Here's the thing: my textbook has an example of using the Gram Schmidt process with an integral. It is stated thus: Let $V = P(R)$ with the inner product $\langle f(x), g(x) \rangle = ...
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1answer
70 views

Is $U=\{(r,0,s)\mid r^2+s^2=0, r,s\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$?

Is $U=\{(r,0,s)\mid r^2+s^2=0, r,s\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$? If I set $r=s=0$, then it shows the zero vector is in $U$. For showing that U is closed under scalar ...
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76 views

Preimage of invariant subspace

Suppose we have a linear map $A \colon V \to V$ on a finite- dimensional vector space, and $W \leq V$ it's invariant subspace. Then we have obviously $\operatorname{Ker} A + W \subseteq A^{-1}(W)$. ...
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68 views

Unique solution to non linear system of equations with boolean coefficients

Say we have a system of $m$ equations of the form: $$a_{11} x_1 + a_{12} x_2 + ... + a_{1n} x_n = p_1$$ $$...$$ $$a_{m1} x_1 + a_{m2} x_2 + ... + a_{mn} x_n = p_m$$ Where the $p_i,x_j \in \mathbb{R}$, ...
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234 views

Find the Wronskian of the Functions [closed]

Find the Wronskian of the functions $f(t)=6e^t\sin{t}$ and $g(t)=e^t\cos(t)$. Simplify your answer. please list out all steps as simple as possible thank you
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2answers
56 views

Is $U=\{(r,s,t)|r,s,t \in \mathbb{R}, -r+3s+2t=0\}$ a subspace of $\mathbb{R}^3$?

Is $U=\{(r,s,t)|r,s,t \in \mathbb{R}, -r+3s+2t=0\}$ a subspace of $\mathbb{R}^3$? So far all I know is that the zero vector is in the subspace. How would I go about checking if it is closed under ...
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64 views

solving equations by the method of substitution

$\dfrac{a}{x}+\dfrac{b}{y}=\dfrac{a}{2}+\dfrac{b}{3},$ $x+1=y$ We have to solve for $x$ and $y$.I have tried to solve for them by finding value of $x$ or $y$ from the second equation and place them ...
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43 views

invariant sub space

So I preparing myself to a test in linear algebra and I scanned the last years test and I reached a question which I do not understand why is it like that. True or false: $ \forall T\colon V ...
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2answers
100 views

distinct eigenvalues implies $\dim(E_{\lambda_i}) = 1$?

as the title states: why does distinct eigenvalues imply that geometric multiplicity of all those eigenvalues is 1? This is used by diagonalization, but it just states it without actually proving ...
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1answer
101 views

Inverse Polynomial in a ring R

I just started working on my Bachelor-Thesis in IT-Security and therefore try to understand the NTRUencryption algorithm. It operates on polynomials in a Ring. My problem is that I don't understand ...
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2answers
97 views

Solving a linear system with complex eigenvalues

I have the system: \begin{equation} x' = \begin{pmatrix}5&10\\-1&-1\end{pmatrix}x \end{equation} The corresponding characteristic equation is: \begin{equation} \lambda^2-4\lambda+5 \\ \implies ...
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1answer
120 views

Every subspace of $\mathbb{R}^n$ is a solution space of a homogeneous system of linear equation.

All solution of $AX = 0$ where $A$ is a $n \times n$ matrix and $X$ is a column vector form a subspace of $\mathbb{R}^n$. All the subspaces of $\mathbb{R}^n$ are of this type. How to prove this ...
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125 views

If $ A^2=0$ , prove that $A$ doesn't neccesarily have a row of zeros

Question $A^2 \in M_{n \times n} (F), A^2=0, n\ge 3$. Prove that it's not true that A necessarily has a row of zeros. Thoughts We thought that the matrix must be nilpotent, but therefore it's main ...
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4answers
105 views

characteristic polynomial and eigenvalues of $T(A)={ A }^{ t }$

Let $V=M_2(\mathbb R)$ and $T(A)={ A }^{ t }$. I was asked to find the characteristic polynomial of $T$ and it's eigenvalues, and finally to say if $T$'s diagonalizable. Is there a way to ...
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3answers
1k views

Dual space and inner/scalar product space

$V$ is vector space of finite dimension. $〈· , ·〉$ is an inner product on $V$.(Field $F$) We set transformation $T \colon V \rightarrow V^*$ as the following: $(T(v))(w) = 〈v , w〉$. Prove that $T$ ...
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2answers
112 views

question from exam: prove the matrix is Invertible

Suppose $d\ge2$, $d$ is an integer, $A$ is an $n \times n$ matrix of integers with each divisible by $d$--that is, each $a_{ij} = 0 \pmod d.$ Prove $I + A$ is an invertible matrix. I tried thinking ...
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63 views

is there a matrix that follows the requirements

Is there a 3 x 3 matrix , name it $A$, that $$A^4 = \left( \begin{array}{rrr} 0& 0& 1\\ 0& 0& 0 \\ 0& 0& 0 \end{array}\right). $$ *sorry I just don't know how to insert ...
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35 views

Find the general forumla for this matrix

A is n x n matrix that in all the places (i, i+1), there is the number 1, and in the other places there are 0 s . (i goes from 1 to n-1) Calculate A^k for k= 1,2,.. and (I-A)^-1 I didn't succeed ...
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247 views

nilpotent transformation properties

I search for a proof for the following theorem but I did not find. I will appreciate a lot if anyone will direct me to a proof or something. Let $V$ be a vector space Let $N$ be a nilpotent matrix. ...
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2answers
89 views

linearly independence of $e^{a_1x},… e^{a_nx}$

$a_1,\ldots,a_n$ are real different numbers. Prove that the functions $e^{a_1x},...,e^{a_nx}$ are linearly independent group in $Fun(R,R)$. My way to try to prove it: I assumed: $b_1e^{a_1x} + ...
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2answers
58 views

Logarithm inequality for vectors

I am trying to prove the following result. Let $d$ be a vector in $\mathbf{R}^{n}$ with $\|d\|_{\infty} < 1$. Then, $$ \sum_{i=1}^{n} \log(1 + d_{i}) \geq \mathbf{1}^{T} d - \frac{\|d\|_{2}^{2}}{2 ...
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2answers
405 views

$n=\dim V$. Then $V=\ker(T^n)\oplus\mathrm{range}(T^n)$

I trying to solve the following problem. The question is from a past exam. Suppose that $V$ is a finite dimensional vector space over a field $K$. Let $T: V\rightarrow V$ be a linear operator. If ...
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70 views

Linear Operators satisfying $S^n=0$ but $S^{n-1}\neq 0$

I need help with part (c). I could do part (a) and part (b).
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220 views

two dimensional linear differential equation with $1$ eigenvector

I have the following linear differential equation: \begin{equation} x' = \begin{pmatrix}3&-4\\1&-1\end{pmatrix}x \end{equation} The corresponding characteristic equation is: \begin{equation} ...
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2answers
806 views

row echelon vs reduced row echelon form

I apologize if this is a very basic question. I understand the difference between the two forms, but i was curious when row echelon from is enough. where is row echelon form used?. Why shouldn't I ...
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4answers
957 views

Is $\{\sin x,\cos x\}$ independent?

Is $\{\sin x,\cos x\}$ linearly independent in $\mathbb{R}^n$? I thought they were not because I can write $\cos x=\sin (x+\pi/2)$. My professor on the other hand said it was independent and his ...
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1answer
245 views

Linear dependence of multivariable functions

It is well known that the Wronskian is a great tool for checking the linear dependence between a set of functions of one variable. Is there a similar way of checking linear dependance between two ...
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1answer
107 views

question from my exam: trivial solution only or more solutions?

$A$ is an $n\times n$ matrix, $n \ge 3$. Assume $A^2 = 0$. Which one is true and which one is false and explain: The linear system $Ax = 0$ has only trivial solution. The linear system $Ax = 0$ ...
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1answer
59 views

Is it possible for a singular matrix to be invariant on this interval?

I'm creating a code that that uses a matrix or matrices as a key. For example, given each string of $n$ letters, construct it into a vector using its position in the alphabet, and multiply it by an ...
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3answers
3k views

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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1answer
111 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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109 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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0answers
109 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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1answer
163 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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3answers
84 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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94 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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2answers
377 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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83 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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365 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
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106 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
330 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
98 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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50 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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3answers
607 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
794 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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2answers
118 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 ...