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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Matrices rank problem

$X\in \text{Mat}_n (\mathbb{R} )$ and $|X|\neq 0$. $X$ has column vectors $X_1,X_2,\ldots ,X_n$. $Y$ is a matrix that consists of column vectors $X_2,X_3,\ldots ,X_n,0$. Let $A=YX^{-1}$ and ...
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16 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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19 views

time and work problem

A man can begin a work at his maximum rate; but afterwards the rate at which he works follows a cyclic pattern. Every 2 hr,it reduces by half but after 8 hrs,it comes back to its maximum level.He can ...
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33 views

Maps from $SO(3)$ to $S_1, S_2$, and $S_1 \times S_2$

I am looking for continuous maps between the special orthogonal group of 3x3 matrices and the unit circle, unit sphere, and their product (S1, S2, S2 x S3, respectively). Any hints as to what I should ...
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12 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
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Eigenvalue of linear operator [on hold]

Let $V=P_3(\mathbb{R})$ and $T$ is a linear operator such that $T(f(x))=xf'(x)+f''(x)-f(2)$. Find the eigenvalues for $T$ and an ordered basis such that $[T]_B$ is a diagonal matrix.
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The Linear Combinations of Two Vectors Fill the Plane Unless _ [Strang P10 1.1.30]

This question is from the 1st chapter of Intro to Linear Algebra, 4th Ed, by Gilbert Strang. So please omit concepts which succeed this question: matrices, rank, REF, nullspace, $Ax=b$, linear ...
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31 views

Vector space clarification

I'm asked to decide if the following are vector spaces. A=$\{f:[0,1] \to \mathbb{R}:\int_0^1|f(x)|dx=0$ $\}$ B= $\{f:[0,1] \to \mathbb{R}:f'(x)+4f(x)=0$ and $f(0)=1 $} C=$\{f:[0,1] \to ...
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33 views

Why is the adjoint a useful concept

I've been reading my linear algebra book, and am now on the section about the adjoint of a linear operator, I get the definition provided and even think I understand the general proof of existence and ...
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45 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
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28 views

You can only take the span of linearly independent vectors?

Ok, this might be a bit trivial but I'm having trouble wrapping my head around my text book. So, to my understanding for ${Span(v_{1},v_{2},..,v_{n})}$ then ${v_{1},v_{2},..,v_{n}}$ must be linearly ...
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21 views

SO(n) is parallelizable

Prove that $SO(n)$ is parallelizable. How would I go about showing this? My supervisor could not help me with this problem, and I am stumped.
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Iteratively find solutions for $AB=H$, where $A$ is known and invertible and $B\geq0$ and $H\geq0$ is unknown?

Iteratively find solutions for $AB=H$, where $A$ is known and invertible and $B\geq0$ and $H\geq0$ is unknown? Can we find one solution by the following procedure? First random pick $H$, then $B$ ...
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24 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
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61 views

Why does a positive definite matrix with a repeated eigenvalue have infinitely many square roots?

So, if we consider a positive definite matrix $A$, (meaning that $A$ is self-adjoint $(Ax,x) > 0$ and also that $A$ has strictly positive eigenvalues) we see right away that since it is self ...
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20 views

Is the inverse of a symmetric positive semidefinite matrix also a symmetric positive semidefinite matrix?

If we let $$S_{++}^n(\mathbb{R})$$ denote the set of all square symmetric positive definite matrix over the real numbers, then is it true if $A\in S_{++}(\mathbb{R}) \implies A^{-1} \in ...
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Linear Representations: Show that no $W^0$ exists.

Given the following linear representation and subrepresentation $W$, show that there exists no $W^0$ such that $\mathbb{R}^2 = W \oplus W^0$. Let $\rho: (\mathbb{Z}, +) \to GL(\mathbb{R}^2)$ be ...
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31 views

Analogy between linear basis and prime factoring

I recall learning that we can define linear systems such that any vector in the system can be represented as a weighted sum of basis vectors, as long as we have 'suitable' definitions for addition and ...
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33 views

Find values so matrix not invertible?

$$ \begin{pmatrix} 2 & 4 & k \\ 1 & 3 & 2 \\ 3 & k & 9 \\ \end{pmatrix} $$ For what values of $k$ is the above matrix not invertible. Need help. Don't know where to ...
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31 views

How are specific linear maps defined?

I'm revising for exams and a question that often crops up is: given a linear map $\mathcal{T}:\;\mathbb{R}^n\to\mathbb{R}^m$, describe how to represent $\mathcal{T}$ as a matrix relative to bases ...
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845 views

Inverse matrix norm under simple conditions

Let $A$ be a real $2\times 2$ matrix such that $\det A=1$, show that $\|A\|=\left\|A^{-1}\right\|$. Any hint would be appreciated, thanks. EDIT: $\|\cdot\|$ is the operator norm ...
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Linear Algebra: show $\sum_{m=1}^{M} a_m x_m = 0$ is a subspace

I have a problem that I can't get my head around. It says that a is any vector in $\mathbb{F}^M$ and to verify (by the three properties of subspaces) that $\sum_{m=1}^{M} a_{m}x_{m} =0$ is a subspace ...
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22 views

Vector Space confusion

For each of the following, I need to decide if it is a vector space over $\mathbb{R}$. (You may assume that the set of all real valued functions on the interval $[-1, 1]$ is a vector space with the ...
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625 views

Orthogonal Projection of a Point onto a Plane

I'm dealing with an exercise that requires I find the orthogonal projection of a given point onto a given plane. I don't want an answer directly for my exercise, I would instead like to understand ...
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27 views

proof of positive semi-definiteness of the precision matrix (inverse of the covariance matrix)

I would like to know how to prove that the inverse of a covariance matrix $\Sigma^{-1}=\Omega$ is positive semi-definite too. My second question can we prove that for any matrix $A\in\cal{M}_{nm}$ we ...
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1answer
72 views

Peculiar Matrix

I came up with this idea recently and I want to go deeper in this, but it has been difficult to me. Hope someone can help me on this. Suppose I have a matrix of order $(n^2-1)\times (n^2-1)$ with ...
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68 views

On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?

I was seeing the proof that $\exp(A)\exp(B)=\exp(A+B)$ on link Show that $ e^{A+B}=e^A e^B$ where uses the hypothesis $AB=BA$? Thanks!
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33 views

Characteristic Polynomial via Induction

I sort of understand where the equation of the Char Poly comes from, but i'm having problems setting up the induction.
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33 views

Determining and enforcing linear dependence

Assuming we have a large set of multi-dimensional vectors (20k vectors, 100 dimensions each). My questions are the following: How can we determine the level of linear dependence of this set? Is ...
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What changes where made on this Gaussian-Elimination?

in the Internet I have found the following use of the Gaussian Elimination method: $z \in \mathbb{R}, \ n\in\mathbb{N}, n \ge 2$ and $\begin{pmatrix} z & 1 & \dots & 1 & 1 \\ 1 & ...
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2answers
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Prove that if $C=A+iB$ is invertible, then so is $A+\lambda B$ for some $\lambda$

I've got a homework question that I've honestly no idea how to tackle. It goes as: Let $A$, $B$ be real $n × n$ matrices such that the complex matrix $C = A + iB$ is invertible. By considering ...
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1answer
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Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
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Transformation Matrix of a function

I have the following: (Note: $V^{*}$ is defined as: $V^{*} = \{ L: V \rightarrow \mathbb{R} | \text{L is linear} \}$) Let $V$ be an $\mathbb{R}$-Vectorspace. Let $\phi \in V^{*} \text{ \ } \{0 \}$ ...
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Cayley-Hamilton Block Matrix

In Q4 d) I understand how to get the expression for A (using Q3), but I don't understand how you can then say that C is of the form described.
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38 views

Find $T(1)$, $T(x)$ and $T(x^{2})$ and $T(ax^2+bx+c)$

Let $$T:P_{3}\rightarrow P_{3}$$ be a linear transformation such that $$T(2x^{2})=2x^{2}+3x, T(\frac{1}{2}x+2)=2x^{2}+4x-3, T(2x^{2}-1)=3x-1.$$ Find $$T(1)$$, $$T(x)$$ and $$T(x^{2})$$ and ...
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$3 \times 3$ matrix with entries $-1$ and $1$

There are $512$ matrix due to $2^9$. Is there a way instead of by hand to find how many of the matrix may equal $1, 2, 3....,$ etc. with the entries being $1$ and $-1$? Thanks in adavance.
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Complex inner product aren't inner products.

An inner product $\langle \cdot,\cdot \rangle$ satisfies the following properties: Let $u$, $v$, and $w$ be vectors and $\alpha$ be a scalar, then: $\langle u+v,w\rangle=\langle u,w\rangle+\langle ...
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69 views

Solution for $x$ with exponents?

I am trying to solve the following, $$7^{(2x+1)} + (2(3)^x) - 56 = 0$$ Should I put the 56 on the other side and get the log of both sides and is there a better way to solve this.
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What is the canonical basis of a dualspace in $\mathbb{R}^3$?

I have the following: Consider the basis $$B := \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \}$$ of the ...
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1answer
12 views

Kinds of finite dimensional inner product spaces

My question is very simple, I'm studying inner product spaces and every Algebra Linear book I read speaks only about vector spaces over $\mathbb C$ or $\mathbb R$, why? are there finite dimensional ...
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37 views

Problem with checking whether $x(t)$ can be a solution of any system of first order homogeneous ODE

I need to find out whether $$x(t) = (3e^t + e^{-t}, e^{2t})$$ can be a solution of the system $$\dot{x} = A x\quad \quad (1)$$, where $A$ is a $2x2$ matrix. I'm not sure of my solution, which is the ...
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A question about polynomials as vectors

The space $P_n(R)$ is very similar to the vector space $R^{n+1}$; indeed one can match one to the other by the pairing $a_nx^n + a_{n−1}x^{n−1} + \ldots + a_1x + a_0$ is equivalent to $(a_n, ...
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26 views

A question about subexponential growth about group

Recall a group $\Gamma$ is said to have subexponential growth if lim sup$|E^{n}|^{1/n}=1$ for every finite subset $E\subset \Gamma. (E^{n}=\{g_{1}g_{2}...g_{n}: g_{i}\in E\}.)$ My question is: Can we ...
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Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
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1answer
21 views

Surjective homomorphism of a linear transformation

In ii) I understand why det is a homomorphism but in the surjective part I cna't get the gist of what is going on. In iii) i'm rather lost as the solution isn't particularly expansive.
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Find bases of matrix without multiplying

This question is related to a solved problem in Gilbert Strang's 'Introduction to Linear Algebra'(Chapter 3,Question 3.6A, Page 190). Q) Find bases and dimensions for all four fundamental ...
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Find linear line of data following gaussian distribution

I have a data that following gaussian distribution $x=\{x_1,x_2...x_n\}, p(x) ~is ~N(\mu,\sigma)$ Now, I want to find a linear line fitting all the data such as $y=ax+b$ The line subject to ...
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32 views

How do I most efficiently find the perpendicular distance from a point to the convex hull of a collection of circles?

I have a collection of one or more line segments for which I know the (x,y) coordinates of the endpoints. The segments may or may not be parallel and may or may not intersect. Each segment endpoint ...
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29 views

Stationary distribution of a “birth-death model” that does not have Markov property

A typical birth-death process is defined such as the probability of going from any state $j$ to any state $i$ is given by: $$ p_{ij}= \begin{cases} b_i & \text {if $j = i+1$} \\ ...
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29 views

minimal polynomial of powers of a matrix

I am wondering, given the minimal polynomial of $A$, what can we say about the minimal polynomial of $A^n$? I know that if we were looking at the roots of the characteristic polynomial ie eigenvalues ...