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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Showing that all eigenvalues are positive numbers,

Let A be an nxn symmetric real matrix with n distinct eigenvalues $\lambda_1 , ... , \lambda_n$. a) Suppose $v^t(Av)$>0 for all v in $R^n$, v$\ne$0. Show that all $\lambda_i$ are positive numbers. ...
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Example of where the sum of a subspace and its orthogonal complement is not the original vector space?

Suppose $\mathbb{F}$ is an arbitrary field and let $W$ be a subspace of $\mathbb{F}^n$. $W^\perp$ can be defined in exactly the same way as in the real case. Show by example that it isn't ...
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Does a basis for an $n$-dimensional vector space have to have $n$ vectors?

For example, for $\mathbb{R}^n$, if I form a basis, do I need at least $n$ vectors in my basis set? In other words, can I form a basis for $\mathbb{R}^n$ by using only $n-1$ or less number of ...
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21 views

Transition matrix question,

In diagonalizing a matrix A, we use a matrix S, which consists of eigenvectors of A. To compute S, we simply take each eigenvector and write it as a linear combination of the standard basis. So if ...
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Inverting change of basis matrices to get back the original coordinate vector

Let $B=\{b_{1},b_{2}\}$ and $C=\{c_{1},c_{2}\}$ be bases for a vector space $V$, and suppose $b_{1}=-2c_{1}+4c_{2}$ and $b_{2}=3c_{1}-6c_{2}$. a. Find the change of coordinates matrix from $B$ to ...
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How do I maximize each value, while having them be as far apart as possible?

I have three values V, S, and A. They sum to 1, and are all greater than 0. How do I maximize each value while having them be as far apart as possible? That is, I'd like V to be clearly greater than ...
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25 views

How do I determine the weight to assign to each bucket?

Someone will answer a series of questions and will mark each important (I), very important (V), or extremely important (E). I'll then match their answers with answers given by everyone else, compute ...
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13 views

Finding change of basis matrix when given two bases as a set of matrices

Find the change of basis matrix between the following bases: $\alpha = \left\{ \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}, \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}, ...
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38 views

Book for Linear Algebra and Matrix

my major is Electrical Engineering and I am new in linear algebra and I need to be familiar with matrix theory deeply because of my research topic which is Image Processing. But, I do not know from ...
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1answer
15 views

Find intersection of non-parallell planes without further assumption on their normals

Finding the intersection line between two planes is basic linear algebra but is it possible to find one formula, without having to dealing with different cases? Example: $$ \left\{ \begin{aligned} ...
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Determining the formula for a linear map

Determine the formula for the following linear map: $L : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $L(1,2) = (0,-1)$ and $L(-1,-1) = (2,1)$. Attempt at solution: On the basis of these examples I ...
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42 views

When is the matrix $\mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T}$ a symmetric matrix?

let $\mathbf{A}\in\mathbb{R}^{m\times n}$ and $\mathbf{x}\in\mathbb{R}^{n\times 1}$. \begin{equation} \mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T} \end{equation} Can we say that ...
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8 views

Sesquilinear Forms: Polarization

This thread is only Q&A.* Given a Hilbert space $\mathcal{H}$. Consider the transforms: $$q[\varphi]:=s(\varphi,\varphi)\quad s(\varphi,\psi):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha ...
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83 views

Books various maths subjects

I am a Civil Engineering student and i am planning on following physics in my career.I want to be ready for the advanced undergraduate courses that i will attend to,so i need to learn Differential ...
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What is pseudospectra of matrix polynomials?

What is pseudospectra of matrix polynomials? Please guide me with some example or some refrence regarding it. Thanks.
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36 views

Basic encoding with math formula

As part of my practice coding, I was given the following problem. Let's say you have the binary string 011100011. One way to encode the string would be to add each digit to the sum of its ...
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26 views

Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
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17 views

polynomial over a field, applied onto a Jordan block

Let $K$ be a field of characteristic $0$, $f \in K[t]$ a polynomial over $K$ and $J \in M_{n,n}(K)$ a Jordan block to an eigenvalue $\lambda \in K$, meaning that $J$ has the shape: $$J = ...
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find a basis of F

This question is related to that one Linear subspace Let $$E=\mathcal{F}(\mathbb{R},\mathbb{R})$$ $$F=\{ f\in E\mid f(x)= e^{3x}(a\cos(2x)+b\sin(2x)),\quad x\in \mathbb{R},a,b\in\mathbb{R} ) ...
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29 views

Finding the kernel, eigenvalues, and eigenvectors of the operator $L(x) := x'' + 3 x' + 4 x$

I want to find the kernel, eigenvalues and eigenvectors of the differential operator: $$L(x)=x''+3x'-4x$$ on the $\Bbb C \space \space \text{vectorspace} \space \space C^{\infty}(\Bbb R)$ as well ...
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18 views

Sum of two kronecker products as a kronecker product

I seek for the following relationship (if there is one so): $$C \otimes D = (A_1 \otimes B_1) + (A_2 \otimes B_2)$$ I would like to obtain $C = f(A_1,A_2)$ (in terms of $A$'s) and $D = g(B_1,B_2)$ ...
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32 views

Linear subspace

Let $$E=\mathcal{F}(\mathbb{R},\mathbb{R})$$ $$F=\{ f\in E\mid f(x)= e^{3x}(a\cos(2x)+b\sin(2x)),\quad x\in \mathbb{R},a,b\in\mathbb{R} ) \}$$ Show that : $F$ is linear subspace of $E$ My ...
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Describe the span of the given vectors geometrically and algebraically

Describe the span of the given vectors geometrically and algebraically: $\pmatrix{1\\0\\-1}$, $\pmatrix{-1\\1\\0}$, $\pmatrix{0\\-1\\1}$. I have figured out that these vectors are linearly dependent ...
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definition of line complex in projective space

In a paper of "R.H.Dye", which you can find here: http://link.springer.com/article/10.1007%2FBF02413785#page-1, I face with a mathematical object "line complex in projective space PG(2n−1,q), I need ...
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35 views

What can we say of eigenvaluesof $L=D-A$?

Given a nonnegative, symmetric, $n\times n$ matrix A, the Laplacian L of A is defined to be $$L=D-A$$ where $D=\operatorname{diag}(d_1,...,d_n)$ and $d_k=\sum_{j=1}^n a_{kj}$; I observe thta $L$ is ...
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30 views

Question on normal matrices

Hello all I was given this question in my linear algebra class which I have tried to solve but to no avail, and I would really appreciate any help. I am given a matrix $ A \in M_{nxn}(C) $ and am ...
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3answers
250 views

If 6x = y+z and 4x = y-z, express z in terms of x

\begin{align} 6x &= y+z\\ 4x &= y-z \end{align} How to express $z$ in terms of $x$? I'm not 100% sure on how to solve in terms of x
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45 views

What is the 2d equivalent of vector multiplication? [on hold]

If two three-dimensional vectors, v1 and v2, are multiplied (i.e. dot product), the result will be a 3x3 matrix. If, instead, there are two three-by-three matricies, what is the corresponding ...
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48 views

Prove H is not a subspace of $R^2$

$H=\{(a+b+2c,ab+c):a,b,c \in R\}$ Please, I need help. I can't solve one single problem on this subject. It just seems finding random counterexamples, I can't see nothing solid. Please help me.
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35 views

Let V be a vector space of dimension n. Prove that no set of n - 1 vectors can span V.

I'm not sure I understand the question. As far as I understand it when it says vector space of dimension n, it signifies that there will be n amount of vectors; right? So basically it wants you to ...
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40 views

How to find a real matrix with complex eigenvalues,

Give a $2 \times 2$ real matrix $A$ with eigenvalues $2+3i$, $2-3i$. I would like hints only. So far, I've been trying get somewhere with $\det[A-(2+3i)I] = 0$ and $\det[A-(2-3i)I] = 0$; which ...
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How are arc components of a spherical system derived?

I am studying a flight dynamics book (see Flight Dynamics by Stengel) and am rusty on spherical coordinates. Commonly, aerospace coordinates use a North/East/Down right-hand system. So $z=-h$, ...
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Orthogonal Procrustes Problem in the Operator Norm

If $A,B\in\mathbb{R}^{n\times r}$ are two matrices, it is fairly easy to see that the solution to the so-called Orthogonal Procrustes Problem $$ \min_{O^TO=Id} \|AO-B\| $$ is given by the polar ...
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33 views

Prove or disprove that the set of polynomials of degree greater than or equal to two, along with the zero polynomial is a vector space

This was disproved by giving the example: $$(x^2)+(1+x-x^2)$$ The result is NOT in the set so it's NOT closed under addiction, so NOT a vector space. But I was looking for some prove that doesn't ...
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23 views

Why is not parity transformation just a rotation?

I'm a bit confused about parity transformations (reflections). A parity operator $\pi$ takes a vector $(x, y, z)$ to $(-x, -y, -z)$. So in a $3$ dimensional space this takes a vector and points it ...
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32 views

Suppose $U=Span\{u_{1}, u_{2} \}$ for $u_{1}, u_{2} \in U$ and $V=Span\{ v1, v2\}$ for $v_{1},v_{2} \in V$. Prove that $U+V=Span\{u1,u2,v1,v2\}$.

This is what I have so far, I don't know if this is where I stop or if there is more to prove? $$U+V = (c_{1}u_{1} + c_{2}u_{2}) + (c_{1}v_{1} + c_{2}v_{2}) = c_{1} (u_{1}+v_{1}) + ...
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If a system of linear equations is inconsistent, what does it mean geometrically?

If we have a system: $$ \left\{ \begin{array}{l} ax+by+z = 1\\ x+aby+z=b\\ x+by+az=1 \end{array} \right. $$ What would be the best way to discuss it? Here's how I started (I used Kronecker–Capelli ...
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35 views

Proving $\phi: V \rightarrow \mathbb{R}^n$ is linear and finding matrix representation of it

Problem: Let $V$ be a $n$-dimensional vectorspace and let $\beta = \left\{v_1, v_2, \ldots, v_n\right\}$ be a basis for $V$. Prove that the coordinate map $\phi_{\beta} : V \rightarrow \mathbb{R}^n$ ...
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23 views

Proof to show that sums of vectors spanning a vector space also span a vector space

Let vectors $v_1, v_2, and v_3$ span a vector space $V$. Show that the vectors $v_1, v_1 + v_2$ and $v_1+ v_2 + v_3$ also span $V$. How would I go about proving this? I understand that I have to show ...
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52 views

Intersection of two planes, how to represent a line?

If we have two planes: $$4x-y+3z-1=0$$ $$x-5y-z-2=0$$ and if we want to find a plane which contains the origin point and the intersection of the two planes given, how do we do it? What my teacher did ...
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Get the camera transformation matrix (Camera pose, not view matrix)

Let's say that I have an object and a camera (its representation) in a 3D world coordinate system. I have the camera pose to see the object (rotation matrix and translation (eye position)). If I apply ...
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Radial Basis Function on 2 dimensional data

I have 2 dimensional point x=(x1,x2). I want to apply Radial Basis Function on this 2D data and transform it to the infinite dimensional space. could any one help me that what will be the new data ...
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What formula would I use for a four factor prioritization method where the factors are summed and ranked?

We are developing a way to prioritize system issues. Our current ranking is 1 - 5, but that becomes rather flat when dealing with a couple hundred issues. In our new method, we have four factors in ...
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21 views

Gradient and invariance under change of basis

I intuitively would be inclined to believe that the gradients $\nabla F_i$ of the components $F_1,\ldots,F_3$ of a vector field $\mathbf{F}:A\subset\mathbb{R}^3\to\mathbb{R}^3$, $\mathbf{F}\in ...
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External operation: binary and unary perhaps???

Consider the following examples from which some definitions are derived: Let us take an element from the set R of real numbers (say, the number 8) and another from the set L of lengths (say, 4m). ...
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Analytical expressions for the orthogonalization of a specific set of vectors

I would like to know whether analytical or closed-form expressions could be obtained for the orthogonalization of a set of vectors in the following setting. Let $x_t$ be a vector indexed as a time ...
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When does a matrix have short vectors in its kernel?

Consider an $n$ by $n$ matrix $M$ whose elements are in $\{0,1\}$, say. Now consider all vectors $v \in \mathbb{Z}^n$. Is there any mathematical property of $M$ which expresses when the kernel of ...
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35 views

An exercise question in Linear Algebra Done Right by Axler [duplicate]

Prove or give a counterexample: if $U_1$, $U_2$, $W$ are subspaces of $V$ such that $V$ = $U_1\oplus W$ and $V = U_2 \oplus W$, then $U_1 = U_2$. I'm a beginner in linear algebra and I'm ...
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Class of matrices for wich $A^T=J-A.$

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix $A$ is symmetric if $$A = A^{\top}.$$ Instead, a matrix of ones or all-ones matrix is a matrix ...