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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Finding a linear map $T:\mathbb R^3 \to \mathbb R^4$ that satisfy conditions about the kernel and image

Find a linear map $T:\mathbb R^3 \to \mathbb R^4$ that satisfy the following conditions (or explain why there can't be such linear map): $\ker(T)=sp\{(1,-1,0)^T,(3,2,-5)^T \} $ ...
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1answer
68 views

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? What is $\dim(\mathbf{X^TX})$?

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? Is this true in general, please? And what is $\dim(\mathbf{X^TX})$, please? Does it equal to $\dim(\mathbf{X})$ ...
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2answers
62 views

Minimal Polynomial of a scalar multiple of a Matrix

I got the following problem: Let $A$ be a square matrix of order $n$ over field $\mathbb{F}$ and let $M_A$ be the minimal polynomial of $A$ of degree $k\in\mathbb{N}$ and let $0\neq c\in\mathbb{F}$ ...
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2answers
86 views

Which of the following statements are true regarding the eigen values of a real matrix

Let $T : \mathbb{R}^n → \mathbb{R}^n$ be a linear transformation of $\mathbb{R}^n$ , where $n ≥ 3$, and let $λ_1, . . . , λ_n ∈ \mathbb{C}$ be the eigenvalues of T. Which of the following statements ...
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1answer
167 views

Consequences when the commutator is a scalar multiple of the identity matrix

I just stumbled over the question below. As to the first, I could easily find out the answer (D) by invoking the commutation relation. But I don't figure out how to solve other two. Could anybody give ...
3
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2answers
134 views

What are the possible values of the determinant of a permutation matrix?

A permutation matrix $A$ is a nonsingular square matrix in which each row has exactly one entry = $1$, the other entries being all zeros. If $A$ is an $n×n$ permutation matrix, what are the possible ...
21
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1answer
280 views

Is a field determined by its family of general linear groups?

Assume that $K,L$ are fields such that there is an isomorphism of groups $\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$ for all $n \in \mathbb{N}$. Does it follow that $K \cong L$? I am also interested in ...
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1answer
51 views

Does countable intersection of linear subspaces with finite codimensions have countable codimension?

Let $E$ be a vector space. Let $F_1, F_2, \dots$ be linear subspaces with finite codimensions in $E$. WLOG we can take the codimensions of the $F_n$'s to be 1. Is it true that ...
4
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2answers
97 views

Connection between even/odd and symmetric/skew symmetric

I read awhile back that the set of continuous real valued functions from $\mathbb{R} \to \mathbb{R} $ has a direct sum decomposition into subspaces of strictly even and odd functions. Any such ...
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0answers
40 views

Jordan canonical form of two commuting matrices [duplicate]

I was asked to prove the following: Suppose that $A$ and $B$ are matrices which commute with each other, that is, suppose that $AB$=$BA$. Show that there exists a matrix $T$ such that both $T^{-1}AT$ ...
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1answer
58 views

How to calculate coordinates of a vector in relation to a basis

Let's say I have a basis and a vector: $$ \mathcal B_1=\{M_1,M_2,M_3,M_4 \} \ \ M_{2\times2}(\mathbb R)\\ v=\begin{pmatrix}a & b\\ c &d \end{pmatrix}$$ Suppose I have numeral values in all ...
3
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0answers
56 views

pullback of density

So I have been reading some differential geometry and they are talking about density's and they claim that a pull back of a density is a density but I only have a partial proof of why this is true. ...
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1answer
65 views

Scaling a cup to have a certain filling volume

I created a cup in Autodesk Inventor using lathe/rotation, ie I defined the profile and rotated it around an axis. I measured it's volume. By using Patch and Sculpt I filled the inner volume(which ...
6
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4answers
164 views

Solve for $X$ in $Y = X^TAX$.

Suppose $$ Y = X^TAX, $$ where $Y$ and $A$ are both known $n\times n$, real, symmetric matrices. The unknown matrix $X$ is restricted to $n\times n$. I think there should be at least one real ...
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2answers
64 views

Is the linear map $\psi: ℂ_2 \to ℂ^2, a+bx+cx^2 \mapsto (a+ib,b-a)$ bijective?

Is the linear map $\psi: ℂ_2 \to ℂ^2, a+bx+cx^2 \mapsto (a+ib,b-a)$ Injective? Surjective? Bijective? I know the definitions of these, but I don't know where to start.
2
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1answer
52 views

When is an orthogonal projection a unitary operator? [duplicate]

Question: When is an orthogonal projection $P: \Bbb C_n \to \Bbb C_n$ a unitary operator? Thoughts: I thought about using the fact that the only eigenvalues for projections are 1 and 0. Don't really ...
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1answer
1k views

Problem with line equation with double equal signs

My excersice is a lot bigger but I really need help with this line equation x-2=4y=z+1 Because it has two equal signs it's throwing me off a bit. Now my question is can I write the equation ...
2
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1answer
195 views

Prove that exist matrix P invertible then A=PB

Let two matrix $A=(a_{ij})_{m\times n}$ and $B=(b_{ij})_{m\times n}$ satisfy $\ker(A)=\ker(B)$ , $\: $($Ax=0\Leftrightarrow Bx=0$) Prove that exist matrix P invertible then A=PB. My tried: ...
4
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2answers
206 views

Determinant of a matrix with generalized binomial coefficients

Let $$ A= \begin{bmatrix}\binom{-1/2}{1}&\binom{-1/2}{0}&0&0&...&0\\ \binom{-1/2}{2}&\binom{-1/2}{1}&\binom{-1/2}{0}&0&&...\\...&&&\binom{-1/2}{0}\\ ...
2
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1answer
49 views

On the structure of a vector bundle

Let $P \rightarrow X$ be a principal $G$-bundle, $\rho: G\rightarrow GL(V)$ and $\sigma: G\rightarrow GL(W)$ be two finite dimensional linear representations of $G$. Let $E=P\times_\rho V$ and ...
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0answers
108 views

How to calculate center coordinates of two reverse arcs in 3D space

Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? ...
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3answers
50 views

Let $S$ be a subset of $V$ . Identify which of the following statements is true:

Let $V$ be a vector space of dimension $d < \infty$, over $\mathbb{R}$. Let $U$ be a vector subspace of $V$ . Let $S$ be a subset of $V$. Identify which of the following statements is true: ...
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2answers
132 views

Inverse of a sum of PSD matrices

I was wondering if anyone knew any techniques to convert the following: $ (A+B+C+..)^{-1} $ where $A,B,C...$ are positive semi-definite (PSD) matrices into a sum of some other function: $ ...
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2answers
560 views

If a matrix commutes with all diagonal matrices, must the matrix itself be diagonal?

I'm new to stackexchange so feel free to correct my style/format/logic etc. The question is this: let's say $A$ is a square matrix of size $n$. I would like to show that $AD = DA$, for any diagonal ...
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1answer
72 views

Eigenvalue of transition matrix

Background: This is an exam problem which I was not able to solve entirely. After the exam, I discussed with other students, almost in vain. The problem: Let $V$ be a $n$-dimensional linear space ...
2
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2answers
62 views

Is there any matrix norm $|| \cdot ||$ such that $||A|| \le ||A||_{\infty} /n$?

Matrix norms are equivalent and can bound each other like some examples on Wikipedia. I was wondering if there is a matrix norm $|| \cdot ||$ that can be upper bounded by $||\cdot ||_{\infty}/n$ ? ...
1
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2answers
81 views

which of the following statements is true regarding characteristics polynomials.

Let $p(x) = a_0 + a_1x + · · · + a_nx^n$ be the characteristic polynomial of a $n × n$ matrix $A$ with entries in $\mathbb{R}$. Then which of the following statements is true? (a) $p(x)$ has no ...
2
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1answer
62 views

Find the dimension of the vector space $R[x]/J$

Let $\mathbb R[x]$ be the ring of polynomials in the indeterminate $x$ over the field of real numbers and let $J$ be the ideal generated by the polynomial $x^3 − x$. Find the dimension of the vector ...
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1answer
34 views

Why $d_xG_{|T_xX}=0\implies d_xG=\lambda_1d_xF_1+\cdots+\lambda_md_xF_m$?

I'm trying to understand this proof of this theorem of a book I'm reading in basic algebraic geometry: Theorem Let $X$ be a closed affine subset and $x\in X$. The restriction of $d_x$ to $I_X(x)$ ...
2
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2answers
95 views

Square vs non-square tensors?

In mathematics, tensors are objects that operates on vector space. In physics or engineering, tensors usually operates on one vector space and its dual space: $V^{*} \times V^{*} \times V^{*} \times ...
4
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0answers
78 views

Identifying the joint distribution from some values of $t \cdot X$

Suppose that $S$ is a subset of $\mathbb{R}^n$ and $X, Y$ are $\mathbb{R}^n$ valued RVs. We already know that $X$ and $Y$ are equidistributed iff $t \cdot X=^d t\cdot Y$ for all $t \in \mathbb{R}^n$. ...
33
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1answer
512 views

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
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3answers
83 views

Why is this true: The only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the identity

Can anyone explain me please how to see this statement: the only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the Identity. how can I prove formally that? or how can I ...
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0answers
32 views

Implementing SVM: Help converting equation into form of another

I'm currently programming a simple linear SVM (Support Vector Machine). For the optimization involved, I need to find a way to convert the equation $\sum\limits_{i=1}^L a_i ...
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0answers
107 views

Find linear independent vectors approximately

Given a set of vectors $a_1,…,a_n∈Z^m$ ($n,m > 10^5$ and $a_i∈Z^+ $ and about $n/2$ elements are zero). How can I approximate orthogonal vectors without using inner product? What vectors' features ...
2
votes
1answer
56 views

If $A$ is diagonalizable in $\mathbb{R}$, is it diagonalizable in $\mathbb{C}$?

I've been wondering, if a matrix $A$ is diagonalizable in $\mathbb{R}$, is it diagonalizable in $\mathbb{C}$? It seems like an obvious yes, but I'm scared I'm missing something. Is my fear grounded ...
2
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0answers
34 views

Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
2
votes
3answers
143 views

Normal matrix and eigenvectors

Let $A \in M_n(\mathbb{C})$ show that if every eigenvector of $A$ is an eigenvector of $A^*$ then $A$ is normal. This question is equivalent to If every eigenvector of $T$ is also an eigenvector of ...
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1answer
30 views

show $(V_1 \cap V_3)+(V_2 \cap V_3)=(V_1+V_2)\cap V_3$ if $ V_1 \subset V_3$

let $V_{1,2,3}$ be Vectorspaces. We want to show the relation above. My attempt: $(V_1 \cap V_3)+(V_2 \cap V_3)=\left \{ v| \exists u_1 \in V_1 , u_2 \in V_2 \cap V_3 , v=u_1+u_2 \right \}=\left \{ ...
2
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0answers
99 views

Optimization - show that linearized feasible set is empty.

I need help in the following problem: Consider the following optimization problem $$ \min_{x_1,x_2}-x_1-x_2\quad\text{s.t.}\quad x_1^2+x_ 2^2-1=0,\quad x_1,x_2\geqslant 0.$$ Show that the ...
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0answers
68 views

Can we list all the orthonormal bases of $C[0,1]$?

Let $C[0,1]$ denote the set of all real valued continuous functions over $[0,1]$. Can we list all the orthonormal bases of $C[0,1]$? In particular my interest to know that does there exist any basis ...
2
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3answers
251 views

solve a linear system using gauss-jordan elimination method

We are asked to solve the following linear system $$x_1-3x_2+x_3=1$$ $$2x_1-x_2-2x_3=2$$ $$x_1+2x_2-3x_3=-1$$ by using gauss-jordan elimination method. The augmented matrix of the linear system is ...
1
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1answer
37 views

which of the following are true regarding the positive definite ness of matrices

Pick out the true statements: (a) Let $A$ be a hermitian $N × N$ positive definite matrix. Then, there exists a hermitian positive definite $N × N$ matrix $B$ such that $B^2 = A$. (b) Let $B$ be a ...
2
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1answer
137 views

Explain why a linear equation has no solution in words

Can I explain why the linear equation $0x_1+0x_2+0x_3+0x_4=8$ has no solution in the following way (This is a question in my homework for elementary linear algebra, $x_i$ are variables.): For any ...
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2answers
273 views

The inverse of a perturbed identity matrix.

Suppose I have a matrix $A$ as following: $$A=\begin{bmatrix}1&a_{12}&\cdots&a_{1n}\\ ...
2
votes
3answers
287 views

Solving matrix equation $AX = B$

I want to solve the matrix equation $AX = B$, where the matrix $A$ and $B$ are given as follows $A = \begin{bmatrix} 0.1375 & 0.0737 & 0.1380 & 0.1169 & 0.1166 \\ ...
5
votes
1answer
95 views

How to compute (and check) this transform matrix?

Background: This is a homework exercise which asks to compute a transform matrix. The answer has been published by our teacher. However, my approach goes a different way and gets a different solution. ...
1
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1answer
94 views

prove that two linear maps over a finite dimensional vector space are conjugate

Let $\alpha$, $\beta$ be linear operators on a finitely-dimensional space $V$ over a field $F$. Let $\gamma=\beta\circ \alpha$ and $\delta=\alpha\circ\beta$. Suppose $\gamma$ is diagonalizable. ...
0
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1answer
63 views

Determine if point is within the bounds of 3 lines

Here is a diagram of what I'm asking: I have the values of p1 through p7. Using these points, how can we determine if p7 is within the bounds of those 3 lines (as pictured in the diagram -- given ...
-1
votes
1answer
71 views

find a common plane which contains two points (NP hard?)

In this problem the coordinates of 4 points are given. $p_{0}=(x_{0},y_{0},z_{0})$, $p_{1}=(x_{1},y_{1},z_{1})$, $p_{2}=(x_{2},y_{2},z_{2})$ and $p_{3}=(x_{3},y_{3},z_{3})$ I need to find the ...