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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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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162 views

Books various maths subjects [closed]

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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2answers
48 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 ...
2
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1answer
88 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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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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2answers
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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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2answers
176 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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48 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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943 views

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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37 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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1answer
40 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
362 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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1answer
56 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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2answers
153 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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4answers
55 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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2answers
50 views

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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28 views

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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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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1answer
110 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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1answer
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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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3answers
41 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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1answer
34 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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4answers
120 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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2answers
217 views

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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1answer
75 views

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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1answer
93 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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37 views

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 ...
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2answers
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Right coordinates of a slanting line when slope is zero and left coordinates never changed after transformation

I have a line in a program I am developing that I want to remove the slant (slope to zero) then get the new coordinates after transformation that removes the slope. This is how the line with the ...
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0answers
38 views

Expanding vector norm into sum

I'm trying to expand a simple Euclidian vector norm into a sum of $x_i$ coefficients, so that for each $i$ term, I can treat everything as coefficients for a quadratic. I think I must have messed up ...
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1answer
88 views

The MU-puzzle from GEB

The MUI system only uses the three letters M,U,and I to make strings. The system has four rules that allow you to make new strings out of existing strings by manipulating them. Rules 1 and 2 lengthen ...
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Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ ...
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2answers
221 views

Reduced row echelon form without introducing fractions at any intermediate stage

How can I reduce this matrix to reduced row echelon form but without using fractions in intermediary steps (I can use them in elementary row operations just not in the results in the matrix) $$ ...
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1answer
111 views

Torsion free abelian group as a $\mathbb Q$-vector space

Let $G$ be a torsion free abelian group of having $n$ number of maximally rationally independent elements $r_{1}, r_{2}, ..., r_{n}$ and assume that $G$ is not finitely generated. Is this correct ...
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proof that every finite matrix has an annihilating polynomial

I don't quite understand the proof my notes gave me. Dimension of $n$ by $n$ matrix is $n^2$. Hence if $k \geq n^2$ then $\mathbf{ \{ I, A, A^2, ..., A^k \} }$ is linearly dependent. So, there exist ...
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2answers
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Proving existence and uniqueness of a matrix,

Let A be nxn with real coefficients and assume that it has n distinct eigenvalues, and all eigenvalues are positive real numbers. Let k $\ge$3 be an odd integer. a) Prove there exists a unique real ...
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184 views

Equivalence of system of nonlinear equations

Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, $b\in\mathbb{R}^n$, $k>0$, and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive function. Consider the system of nonlinear ...
5
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1answer
66 views

Does it make sense to talk about complex matrices over the field of real numbers, R?

I don't see an issue with considering a vector space of complex matrices over R -- addition of matrices makes sense, but scalar multiplication will be done with real numbers. But I wanted to ask, ...
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1answer
54 views

Determinant proof question.

Using determinants, prove that if $A_1,A_2,...,A_m$ are invertible $nxn$ matrices, where $m$ is a positive integer, then $A_1A_2...A_m$ is an invertible matrix. Need help starting the proof. Do I ...
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42 views

Determinant Question.

Show that if $A=\begin{bmatrix}a & b\\c & d\end{bmatrix}$, then $\det(A)=\frac{1}{2}\det\left(\begin{bmatrix}1 & 1\\tr(A^2) & (tr(A))^2\end{bmatrix}\right)$. I tried finding the ...
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1answer
51 views

Determining kernel and image of linear map

Problem: Which of the following maps are linear? Determine the kernel and the image of the linear maps and check the dimension theorem. Which maps are isomorphisms? 1) $L_1: \mathbb{R} \rightarrow ...
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2answers
29 views

Matrix multiplication and determinant question

Show that if $\det(\begin{bmatrix}b & c\\a & b\end{bmatrix})=0$ with $A=\begin{bmatrix}a & a\\b & b\end{bmatrix}$ and $B=\begin{bmatrix}b & b\\c & c\end{bmatrix}$ then ...
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2answers
63 views

What is the correct way to write this matrix equation?

Given an $n \times m$ matrix $X$ and $m \times m$ matrix $A$, I would like to define the vector $y$ as $$y_i = X_{i,*} A (X_{i,*})^T$$ where $X_{i,*}$ is the $i$th row of $X$. Is there a simpler ...
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58 views

Block Matrix Zero Determinant Implication?

Recently I've been working with a number of square (order of 2n) matrices whose determinants are zero. That is, $$\det\begin{bmatrix}A&B\\C &D \end{bmatrix} = 0$$ where each of A,B,C, and D ...
4
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3answers
169 views

Linear dependence of these functions?

How can I check if these three functions (which belong to vector space $R^R$) are linearly dependent: $$e^{2x}, e^{3x}, x$$ If I take $\alpha, \beta, \gamma ∈ R$ and write the linear combination as: ...