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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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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1answer
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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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
43 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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13 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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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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24 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
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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3answers
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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4answers
54 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
26 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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7 views

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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14 views

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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22 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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1answer
26 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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16 views

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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43 views

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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1answer
36 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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35 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
34 views

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
21 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
29 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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4answers
508 views

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
27 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
79 views

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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45 views

Not sure if my Math Operation right or wrong. [on hold]

Hello guys I'm stuck with this. I'm not sure if my math operation right or wrong(Sorry for my bad English) Equation located here http://1.1m.yt/Cl3lGSbEn.png m=z=>>0?(z1-z2%50):(z1-z2%50)-50
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4answers
40 views

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
32 views

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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31 views
+50

Equivalence of system of nonlinear equations

Let $A\in\mathbb{R}^{n\times n}$ be a semi-positive definite, $b\in\mathbb{R}^n$ and $k>0$. Consider the system of nonlinear equations $$ (1) \quad Ax=-k\frac{x}{g(x)}-b. $$ Let $A^+$ be the ...
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1answer
54 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
26 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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2answers
37 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
29 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
25 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 ...
2
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2answers
42 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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2answers
30 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 ...
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152 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: ...
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0answers
9 views

How to build an 2-D polynomial from 1-D orthogonal polynomials

I have an set of orthogonal polynomials such as I want to build an 2D polynomial following the equation $$P_k(x,y)=P_k(x)P_k(y)$$ where $k=1..4, (x,y) \in [-1, 1]^2$ Based on given $P_n(x)$ as ...
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1answer
26 views

Which polynomial has similar properties with Legendre?

I am looking for an kind of polynomial such as Legendre properties that polynomial sequence of orthogonal polynomials such as bellow image. Could you suggest to me one polynomial? Is B-spline correct? ...
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1answer
12 views

dimension of the quotient of a bialgebra

I am stuck in a proof of a lemma that I am in need of. The situation is as follows: Let $k$ be a field and $A$ and $B$ two finite-dimensional $k$-bialgebras, where the dimension of $A$ is a prime ...
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3answers
49 views

Let $v_{1}=(1,-2,3),v_{2}=(0,-1,2)$. Enlarge $\{v_{1},v_{2}\}$ to a basis for $\mathbb{R}^3$.

For instance, let $v_{1}=(1,-2,3),v_{2}=(0,-1,2)$. The set $\{v_{1},v_{2}\}$ is linearly independent. Enlarging $\{v_{1},v_{2}\}$ to a basis for $\mathbb{R}^3$ I simply form a matrix using ...
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1answer
12 views

How does this form of Poincare's inequality for self-adjoint matrices hold?

I'm reading "Introduction to Matrix Analysis and Applications" by Hiai and Petz, and they state Theorem 1.26 ("Poincare's Inequality") as follows: Let $A\in B(H)$ be a self-adjoint operator with ...
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1answer
22 views

Variance of subset vs total variance

Is it true that the variance of subset is smaller than variance of the total set? Given each element in the set is a N-dimensional vector, and the distance is defined as Euclidean distance. Variance ...
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0answers
21 views

Rank of the sum of two rank 1 matrices, proof check

Claim: $(\forall u\in \mathbb{R}^2)$ $(\nexists(\delta,v)\in(\mathbb{R}, \mathbb{R}^2))$ such that $uu'+vv'=\delta \begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$. That is, for any vector $u$ of ...
2
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0answers
15 views

Put, in matrix form: $t_i=\sum_{j=1}^n \frac{w_j-w_i}{1+e^{-(x_i-x_j)}}$, $\forall i=1,2,…,n$

I have the set of equalities $$t_i=\sum_{j=1}^n \frac{w_j-w_i}{1+e^{-(x_i-x_j)}}, \ \ \forall i=1,2,...,n$$ and I try to write them in a more concise form. I tried to do so: $$t_i=\sum_{j=1}^n ...
0
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1answer
30 views

Solutions to the equation $a + b - ab/t = t/2$

$$a + b - ab/t = t/2$$ Where $0 < a < b < t$, $a,b,t \in \mathbb{N}$ and t is even, ie $t\mod2 = 0$ What are the possible values for a, b for a given t? For example, if t = 1000, then a = ...
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0answers
25 views

is this conclusion true or false?

Let $\mathcal{A}$ be a factor Von Neumann algebra and $\Phi$ is a map on $\mathcal{A}$ which is injective and surjective and $\Phi(0)=0$. If $A, B, C \in \mathcal{A}$ and ...