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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1answer
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Keeping the arc length constant between points in a spiral

I'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is ...
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2answers
59 views

Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
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3answers
60 views

How to solve this Quadratic Word problem?

This is the word problem. If they work together, John and Vince can finish their project in Biology in two days. If they worked individually, it will take John three days longer than Vince to ...
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0answers
33 views

Linear Algebra, Quadric form

Question from an exercise $V$ is a vector space over the field F with $charF\neq2$. If $\varphi,\psi\in V^{\vee}$ are linear functionals, we will define $\varphi\cdot\psi:V\rightarrow F$ ...
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relative sign in hodge * of tensor product

For $\delta_i \in \bigwedge^{k_i}W_i^*$, $i=1,2$, the Hodge $*$-operator of $\delta_1 \otimes \delta_2$ is given by $$ *(\delta_1 \otimes \delta_2)=(-1)^{k_1k_2}(*_1\delta_1) \otimes(*_2 \delta_2)$$ ...
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1answer
74 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
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1answer
24 views

Integer QR decomposition

Let $A$ be a real $m\times n$ matrix and $A=QR$ be the QR decomposition of $A$. For what integer elements of $A$ do $Q$ and $R$ have integer elements? I think there are two approaches: Constructing ...
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1answer
29 views

What is the number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My guess ...
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2answers
26 views

Invertible Linear Maps Proof [on hold]

1) Suppose $V$ is finite dimensional and $S$, $T$, $U \in L(V)$ and $STU = I$. Show $T$ is invertible and $T^{-1} = US$. 2) Suppose $V$ is finite dimensional and $R$, $S$, $T \in L(V)$ are such that ...
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1answer
13 views

Linear Operators Injectivity and Surjectivity

Suppose T $\in L(P(R))$ is such that T is injective and deg Tp $\leq$ deg p for every nonzero polynomial p $\in P(R)$. Prove that T is surjective and that deg Tp = deg p for every nonzero p $\in ...
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20 views

matrix function onto and 1-1

I have just started a linear algebra paper and we are doing 1-1 and onto functions. I understand in theory what they mean, I just don't know how to prove them. For example: Define $f: ...
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1answer
51 views

Counterexample of $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$

I know that $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$, where $T^{*}$ means the adjoint operator of a linear operator $T$, holds when the domain of $T$ is finite-dimensional. However, the proof uses ...
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1answer
53 views

Invertibility of $I-AB$ [duplicate]

I got a question in linear algebra: 1) Let A and B be $n\times n$ matrices. If $I - AB$ is an invertible matrix, then prove that $I - BA$ is invertible. Can someone tell me how to solve this ...
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1answer
28 views

How to write this system in the form Ax=b

Given the following system of N equations with N unknowns, with $\lambda$ known and the $a_{ij}$'s also known entries of an m*n matrix A. How would you express the system in the form A x=b? x is of ...
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5answers
80 views

If $A^2$ is the zero matrix, show that $A$ is linearly dependent?

The original question was show that $0$ is an eigenvalue for the matrix $A$. This was a straightforward practice of righthand multiplication of $Ax = \lambda x \Rightarrow AAx = A \lambda x ...
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2answers
37 views

Nullity of a matrix

How would I do D3 and D4? I am completely lost
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0answers
27 views

Proof of Strong Duality via Farkas Lemma

I am trying to prove what is often titled the strong duality theorem. There is a hint in the book that I'm following, and I want to follow the method they have outlined for me. I will outline the ...
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3answers
101 views

Strange proof of Schwarz Inequality with Pythagorean Theorem

Does anyone know what is going on in this proof of the Schwarz inequality? Most importantly: how can one assume that $c^2\leqq \|A\|^2$, or later on, that $c^2\|B\| \leqq \|A\|^2$? This would imply ...
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0answers
48 views

Why does the Fourier transform include the base of the natural logarithm, the square root of -1 and $\pi$?

The formula itself, as a vector of summations of products of the original coefficients with some weight, itself a function the original and transformed coefficient indices, is not a hard pattern to ...
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3answers
38 views

Show that the image of a linear transformation is equal to the kernel

Let $\phi$ be a linear transformation such that $\phi: V\to V$ We are given the following facts: $\dim(V) = 8$ $\dim(\mathrm{Im}(\phi)) = 4$ $\phi\circ\phi=0$ Show that $\mathrm{Im}(\phi) = \ker ...
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2answers
98 views

Prove that if the sum of each row of A equals s, then s is an eigenvalue of A. [duplicate]

Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s$. Show that $s$ is an eigenvalue of $A$. [Hint: Find an eigenvector] My attempt: By definition: ...
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0answers
25 views

Is a given vector in Range of a matrix?

How do I show that a given vector is in range of a matrix without solving for variables. Is there are way of doing this with null space?
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2answers
66 views

prove the following $(A^t)^{-1}=(A^{-1})^t$

$(A^t)^{-1}=(A^{-1})^t$ Proof: $(A^{-1})^{t}*A^T=(A*A^{-1})^t=I$ How to continue from here?
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1answer
21 views

Logarithmic spiral appears inverted

I'm learning to code the equation for a logarithmic spiral for a graphics visualization. However, it appears to be inverted with the radius getting smaller (rather than larger) toward the outside of ...
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2answers
40 views

Subspace Equations

I'm studying the book "Finite Dimensional Vector Spaces" by Paul Halmos. I'm doing q5 from $\S 12$ Dimension of a Subspace, in chapter $1$. I'm not all that used to L.A. proofs, so I'm looking for ...
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1answer
27 views

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
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0answers
25 views

Iterative methods for solving a linear equation system

There are several methods known for solving a linear equation system Ax = b (like Jacobi or Gauss-Seidel) by iterating $x_{n+1}=Mx_n+c$ with a matrix M, for which some norm is smaller than 1. But ...
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1answer
41 views

Question about proving symmetric matrices are diagonalizable

Definition : If a n by n matrix $A$ is orthogonally congruent to another matrix $B$, then there exist an orthogonal matrix $C$ such that $$A = C^{-1}BC$$ Theorem: If $A$ is symmetric, then $A$ is ...
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2answers
44 views

$\ker(A)=\text{Im}(A^*)^\perp$

How do I show that $\ker(A)=\text{Im}(A^*)^\perp$ for any square matrix $A$. I have done this problem before with the linear operator $T$ on a hermitian space but I can't seem to apply what I have ...
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1answer
26 views

Will a 2 by 2 quadratic form be negative definitive if it has repeated eigenvalues which are negative?

Say we have the quadratic form $$ f = x^T Q x \\ Q = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}$$ which has repeated eigenvalues $\lambda = -1$. Will the quadratic form be negative ...
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3answers
59 views

Whether $\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$ is true

Suppose we have k positive numbers: $\alpha_1, \alpha_2, ..., \alpha_k$, for any number $\beta>0$, is $$\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$$ ...
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0answers
37 views

Defining an inner abstract vector space

Since an inner product space is an abstract vector space with an additional structure called an inner product, and this additional structure is a component wise operation that associates each pair of ...
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2answers
43 views

Linear Algebra, Quadric Form, Bilinear Form

I have a question from an exercise. So $V$ is a vector space with quadric form $q:V\to \mathbb{R}$ . I have to prove that if the exists $u$, $v$ in $V$ such that $q(v)>0$ and $q(u)<0$ then ...
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1answer
40 views

Prove: Square Matrix Can Be Written As A Sum Of A Symmetric And Skew-Symmetric Matrices

Let $C^{n \times n}$ be a square matrix. Prove that $$C=\frac{1}{2}(C+C^t)+\frac{1}{2}(C-C^t)$$ What I have manage so far is: a. Let $S$ be a Symmetric Matrix so $S=C+C^t$ b. Let $N$ be a ...
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1answer
65 views

Why Trace and the main diagonal of a matrix are distinguished

Let $A$ be a square $n \times n$ over a field (say $\mathbb{R}$ or $\mathbb{C}$). As we know, the main diagonal $(a_{1,1},...,a_{n,n})$ is important in linear algebra while the off-diagonal is far ...
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25 views

solving equation also involving unknown matrix in trace

Given two real $m$ x $k$ matrices $A_1$ and $B_1$ and two $k$ x $k$ real matrices $A_2$ and $B_2$ I want to solve the following equation for $Q$. $Q$ is an orthogonal matrix, i.e. $Q^TQ=I$. ...
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24 views

Does there exist a pair $(i,j)$ such that $x_{i}(1-x_{j})$ and $x_{j}(1-x_{i})$ are not greater than $\frac{1}{4\cos^{2}\frac{\pi}{n+1}}$?

Let $x_{1},x_{2},\ldots,x_{n}\in[0,1]$. Do there always exist $i,j$, $1\le i<j\le n$, such that both $x_{i}(1-x_{j})$ and $x_{j}(1-x_{i})$ do not exceed $\frac{1}{4\cos^{2}\frac{\pi}{n+1}}$?
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Proof that quantum relative entropy is $\leq$ 0 using Klein's inequality for positive semi-definite operators

I was asked to prove that $S(\rho) \leq - {\rm Tr} \left[ \rho \log \tau \right] $ where $\rho, \tau$ are density operators on a finite dimensional complex inner product space and $S(\rho)$ is the von ...
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1answer
20 views

Find angle of incomplete rotation matrix

I'd like to find the angle of rotation of the following matrix $A= \begin{bmatrix} -\frac{1}{3} & \ast & \ast \\ \ast & -\frac{1}{3} & \ast \\ \ast & \ast & -\frac{1}{3} ...
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45 views

Why should I care about eigenvectors/eigenvalues [duplicate]

I've been studying pattern recognition/machine learning and the theory behind it for some time now and I notice that I find myself seeing the same things over and over again, yet without fully ...
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1answer
27 views

Differential function as a linear map.

Consider the Linear Map $T\colon P_{3}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R})$ given by the set of infintely differentiable functions 1) $T(f)\colon x \mapsto xf'(x)$ , Prove that $T$ is ...
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1answer
24 views

Orthogonal projector onto an eigenspace of a self-adjoint operator

Suppose that $A$ is a self-adjoint linear operator on a Euclidean finite-dimensional space $V$. Is it true that any orthogonal projector $P_\lambda$ onto an eigenspace of $A$ can be represented as a ...
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1answer
48 views

Basis for space of matrices in $\mathbb M_2(\mathbb R)$

Given that $G=\left\{ \left(\begin{array}{cc} a & -a\\ b & c \end{array}\right):a,b,c\in\mathbb{R}\right\} $ and $H=\left\{ \left(\begin{array}{cc} x & y\\ z & -z ...
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1answer
25 views

shortest point on a line segment from point out side the line

from the above pic I found the value x from line (p1,p2) and point a using y=mx+b and imaginary red line which is perpendicular to black line having slope -1/m and the intersecting point x. the ...
2
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2answers
52 views

Decompose a real symmetric matrix

Prove that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $A$ ...
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1answer
24 views

How to prove that the linear independency of $f_{i}$ from $C^{n-1}\left [ a, b \right ]$ to $C\left [ a, b \right ]$

How to prove that if $f_{1}, f_{2}, ..., f_{n}$ are linearly independent in $C^{n-1}\left [ a, b \right ]$, then they will also be linearly independent in $C\left [ a, b \right ]$.
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3answers
318 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
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3answers
54 views

The $R^x$ notation?

I am repeating some linear algebra and I can't remember how to read statemeents like this: $$ T: R^4 \rightarrow R^2 $$ There is a transformation of some sort, right? But What does the 4 and the 2 ...
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1answer
22 views

What is the Jacobian of the following function

Consider a function F: $R^n \to R^n$ defined by $$f(u) = A*u*(n+1)+\lambda *B$$ Where A is a tridiagonal n-by-n matrix with -2 on the main diagonal and 1 on the off diagonals. B = $\begin{pmatrix} { ...
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1answer
24 views

Linear Maps from a finite space to an infinite space

Suppose V is finite dimensional with dim V > 0. Prove that if W is infinite dimensional then $L(V, W)$ is infinite dimensional. Help? I really have no idea how to go about this one? I'm assuming I ...