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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How to construct a symplectic basis from vectors in phase-space?

So I have a set of data from a phase-space for example in $\mathbb{R}^{2n}$ and I want to build a symplectic basis for this space i.e. if A is my symplectic basis then, $ A^T \mathbb{J}_{2n} A = ...
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3answers
35 views

Line of intersection of two planes

So, this question is more like two mini-questions that are subsets of a single regular-sized question. Say I have two planes: $x-z=1$ and $y+2z=3$. I'm trying to find their line of intersection. a. ...
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1answer
18 views

Prove basis exists for a weird transformation

Suppose that $S : \Bbb{R}^2 → \Bbb{R}^2$ is a linear transformation such that $S^2 = S, S\ne 0 $ and $ S \ne I$ Prove that there is a basis for $\Bbb{R}^2$ with respect to which the matrix $A_S$ of ...
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1answer
38 views

Change of Eigenvectors by Hadamard product.

Please forgive me if my question is not clear and appropriate but please do not give negative marking as I am trying very hard to answer this question; thank you. My question is related to the ...
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1answer
54 views

$\psi ( \frac {\pi}{2}, \frac {\pi}{6})$ and calculating problems?

I ran into a problem, $u=\psi (x,t)$ be a solution of partial deferential equation with following condition on boundary, how we reach the value of $\psi ( \frac {\pi}{2}, \frac {\pi}{6})$? ...
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76 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in Polar coordinate $ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial u} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 u} {\partial \theta^2}$ with ...
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0answers
31 views

Algebraic multiplicity of an eigenvalue $λ$

I was going through a question posed on the expression for algebraic multiplicity of an eigen value $\lambda$ on this page : Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is ...
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14 views

How to get the projection matrix from coordinate/transformation?

I would like to compare my results with the groundtruth provided by a dataset. For each frame (image) in the groundthruth, I have a projection matrix. For example (for the 0th frame): ...
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1answer
26 views

Determinant using Row and Column operations/expansions

We are asked to show that: $$ \det\left[\begin{array}{rrr} 2 & 3 & 7 & 1 & 3\\ 2 & 3 & 7 & 1 & 5\\ 2 & 3 & 6 & 1 & 9\\ 4 & 6 ...
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3answers
28 views

Help understanding Vector Space Axioms

I am having a difficulty trying to understand an axiom regarding vector spaces. There exists an element $0$ in $V$ such that $x + 0 = x$ for each $x\in \mathbb{R}$ Two examples, that I don't ...
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1answer
33 views

What are the names of these variations on the transpose of a matrix and symmetric matrices?

Is there a name for the operator that reflects a matrix over the diagonal running from the top-right to the bottom-left? For the moment, define this reflection of a matrix $A$ as $A^*$. Is there a ...
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1answer
39 views

Determine if the set of all $f \in C(\mathbb{R})$ such that $f(1/2)$ is a rational number is a subspace of $C(\mathbb{R})$

Let $C(\mathbb{R})$ denote the vector space over $\mathbb{R}$ of all continuous functions on $\mathbb{R}$. Determine if the set of all $f \in C(\mathbb{R})$ such that $f(1/2)$ is a rational number is ...
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3answers
19 views

Characteristic polynomials and finding a certain Transformation.

So I need help understanding a solution on a practice quiz I have. Here is the question, the above results are explained below. So I get the logic behind how they set up to find the solution by ...
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2answers
22 views

How do you compute an expression containing complex numbers with large powers?

$$(\frac{-\sqrt{3}}{2}+\frac{1}{2}i)^{123}=i$$ $$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$ So I have these equations with the answers ...
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0answers
31 views

Confusion about finding matrix of Linear Transform w.r.t to different bases

I have come across two questions about matrices and changes of bases. They seem to be the same question, but require different approaches. I can't figure out why. First question can be found at: ...
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12 views

Linear algebra and a linearly dependent matrix [on hold]

I have a 9x5 matrix that I am supposed to solve. There are two rows that are linearly dependent of each other. I was told I need to average out the two observations in the b vector. This will allow me ...
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0answers
16 views

Proving existence of Hermitian Adjoint in unusual way

For a map $T:V\rightarrow V$, we define the Hermitian adjoint to be the unique $T^*:V\rightarrow V$ such that $\langle Tu,v\rangle = \langle u, T^*v\rangle$. There are two things I'm required to ...
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1answer
24 views

Prove vector spaces are isomorphisms

Suppose that $V$ is a finite-dimensional vector space over $\Bbb{F}$ and that $T : V → V$ is an isomorphism. Prove that if $S : V → V$ is also a linear transformation and $ST$ is an isomorphism, then ...
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19 views

Shortest distance and Cross Product [on hold]

Show that the shortest distance from a point P to the line through Po with direction vector d is $$ ||P_oP \times d||/||d||$$. I need help writing the proof for this. So far I have: let $ ...
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1answer
19 views

Prove linear independence and spans with linear maps

Suppose that $V,W$ are vector spaces over $\Bbb{F}$ and that $T : V → W$ is a linear transformation. (a) Suppose that $T$ is one-to-one, and that $\{v_1, · · · , v_n\}$ is linearly independent in $V$ ...
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2answers
23 views

Row Reduce Echelon Form on 3x4 Matrix

I understand the rules for RREF are: 1) Each leading entry must be a 1 in each row 2) Each leading entry's column must be 0's other than the leading entry 3) In stair case order, the next element of ...
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36 views

Looking for help on linear algebra theorem

Hello I saw a theorem in my class, and I tried to go back to my book to review it , and it wasn't covered in the book. Also, I unfortunately didn't catch if there was a specific name or title to it. I ...
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3answers
46 views

Showing linear independence of $\{5, e^{ax}, e^{bx}\}$

The question is as above. I have not learnt about the Wronskian or the likes. I suppose that I'll have to do something with differentiation or trying different x values. EDIT: $a$ and $b$ are ...
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1answer
41 views

What is known about optimization of spectral properties of matrices over finite fields?

[I am solving the characteristic polynomial over complex numbers but since the matrices are symmetric all eigenvalues are real] Like for symmetric $d-$regular matrices over 0/1 or 0/1/-1 what are ...
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23 views

Reflection about line? Linear Algebra.

So the question is asking to find the transformation in $R^2$ for a reflection around a standard y=mx+b line. I understand how to do the change of basis part. I don't understand the following part of ...
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2answers
21 views

LU factorization less efficient than Gauss elimination if only used for one {b} vector?

Here is my thought process: Gauss elimination requires ~(2n^3)/3 flops for forward elimination and then ~n^2 flops for back substitution. LU factorization requires a forward elimination to obtain the ...
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0answers
12 views

What is or how do you get the rotational matrix of 4-D vector onto the xyz-space?

which would make the 4-D component 0. To be honest I'm not really sure how 4-D rotations work. I know about the simple rotations but not the mechanism in how it rotates, and I'm not sure whether to ...
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1answer
17 views

Explain $(\|Mx\|_2)^2 = (M^Tx)^T(M^Tx) $ (positive definite, positive semi definite)

Would really appreciate if someone can explain: $$ (\|Mx\|_2)^2 = (M^Tx)^T(M^Tx) $$ can't get my head round with this.
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2answers
57 views

Prove that the set of bases is linearly independent

Suppose that $W$ and $W'$ are subspaces of the vector space $V$ with the property that $W\cap W'=\{0\}$, and suppose that $\beta$ is a basis for $W$ and $\beta'$ is a basis for $W'$. Prove that the ...
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29 views

Proof theorem of Lie's Algebra [on hold]

I need help with a proof this theorems, anyone could be help? I need a proof this theorems. Corollary of Lie's Theorem: Let $L$ be a solvable subalgebra of $\text{gl}(V)$, $\dim V = n$ (finite). ...
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0answers
19 views

Find rank of parametered matrix

Today, I've got task: Find all values of parameter a, when rank of matrix M equals 2, where matrix M is 3x3 and has some dependencies on a, for example: $$\begin{pmatrix} 1 & a & a^2-1\\ 1-a ...
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2answers
35 views

Euclidean distance and dot product

I've been reading that the Euclidean distance between two points, and the dot product of the two points, are related. Specifically, the Euclidean distance is equal to the square root of the dot ...
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1answer
27 views

Solve z in an expression involving complex conjugates.

Solve for z, and give your answer in the form a+bi. $$\overline{z+2-2i} = {2z + 5 - 7i}$$ I know fully understand the concept of complex numbers and complex conjugates. I've found that the answer is ...
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1answer
12 views

How to validate the basis of the space of solutions of a system of linear equations

Problem Solve the following nonhomogeneous system of linear equations: $$ \begin{aligned} x_{1} + 2 x_{2} + 3 x_{3} + x_{4} &= 4,\\ 2 x_{1} + 2 x_{2} + 3 x_{3} + x_{4} &= 5,\\ 3 x_{1} + 3 ...
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1answer
32 views

Finding a basis for a subspace. Do i always need to test linearly independence?

Where the subspace is contained in {[5r-3s;2r;0;-4s] is an element of R^4: r and s are scalars} The generating set that can make up all of the input is {[5;2;0;0], [-3;0;0;-4]} This is only a ...
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2answers
43 views

Inverting matrix multiplication “and” representing with a smaller sized matrix

Consider I have a vector $A=[a_0 \ \ a_1]$ and a random binary matrix $B$ which is $2\times 2$. I compute $C=A\cdot B$. My question is: " Can one compute $B$ Given $C$ and $A$? " Note: By binary ...
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0answers
11 views

Find ellipsoid axes from inertia tensor

The inertia tensor is defined as $I_{ij} = \sum_k \, m_k(r_i^2\mathbf{1}-x_{ki}\,x_{kj}) $, where $\mathbf{1}$ is the identity matrix and $r_i=(x_i,y_i,z_i)$. I know that for an ellipsoid with ...
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1answer
82 views

If A is a square matrix, and A^2 = 0 then A=0 Prove true or provide a counter example?

This is a proof question and I am not sure how to prove it. It is obviously true if you start with A = 0 and square it. I was thinking: If $ A^2 = 0 $ then $ A A = 0 $ $ A A A^{-1} = 0 A^{-1}$ ...
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0answers
16 views

Casting out of linearly dependent vectors from a set

Lemma: Let $V$ be a vector space and let $S=\{v_1,\ldots,v_n\}$ be a linearly dependent subset of $V$ and let $U={\rm Span}(S)$. Then there exists an index $1\le h\le n$ such that if $T=S\,\backslash ...
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0answers
18 views

Why matrix can have only two possible Jordan Canonical Forms

If matrix $A = \left(\begin{array}{ccc}1 & 1 & 1 \\0 & 1 & 0 \\0 & 0 & 3\end{array}\right)$. Why is the case that the following two matrices, $B,C$ are the only two possible ...
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1answer
35 views

Are rotations the result of composing two reflections?(Linear Algebra)

I mean, is it true that every rotation matrix is the result of multypling one reflection matrix by another? If the answer is yes, how do I prove it? And what are the reflection matrices I can use to ...
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16 views

Minimizing l2-norm of convolution (Perron-Frobenius theorem)

I need to minimize the $||\mathbf{h}*\mathbf{x}||_2$, where $\mathbf{h}$ is a given non-negative vector, and $\mathbf{x}$ should be a compactly supported non-negative vector. In the matrix form, this ...
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1answer
18 views

Finding Jordan Canonical form for 3x3 matrix

I was looking at http://www.math.hkbu.edu.hk/~zeng/Teaching/math3407/Jordan_Form.pdf (section 2) $A =\left(\begin{array}{ccc}4 & 0 & 1 \\2 & 3 & 2 \\1 & 0 & ...
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3answers
33 views

Eigen values of a transpose operator

Let $T$ be linear operator on $M_{nxn}(R)$ defined by $T(A)=A^t.$ Then $\pm 1$ is the only eigen value. My try : Let $n=2,$ then $[T]_{\beta}$ = $ \begin{pmatrix} a & 0 & 0 & 0 \\ ...
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0answers
28 views

Rank of a matrix with parameters

I have the following matrix: $$\begin{pmatrix} b+3 & a & 4 & -2b-1\\ b & -3 & 5 & -6\\ -1 & 1 & 2a+1 & 1-a \end{pmatrix}$$ How can I determine the rank for ...
2
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0answers
26 views

Decompose a vector space into invariant subspaces?

Consider the following proposition: Suppose $V$ is a finite dimensional vector space over a field $F$, and $K/F$ is a finite Galois extension with Galois group $G$. If $V$ has a $(K,K)$ bimodule ...
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1answer
32 views

Proof linear transformation

I'm wondering how we could prove : $ \text{Let} \ E, F, G, \text{be three finite dimensional,} \ \mathbb{K}- \text{vectorspaces}, \\ L \in \ \mathbb{L}(E,G) \ (\text{linear transformation of E into ...
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1answer
18 views

Cross product matrix

I accidentally found that curl can be represented as a matrix, and conformed it on wiki: $$[\nabla ]_{\times}=\begin{bmatrix} 0 & -\partial_z & \partial_y \\ \partial_z & 0 & ...
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1answer
29 views

How to visualize a line integral

I was studying for multivariable calculus and I came across the line integral section. Visually, I perfectly understand why $\int_{t_1}^{t_2} f(x(t),y(t))s'(t)\,dt$ computes the area under $f(x,y)$ ...
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15 views

Matrix induced norm by vector norm defined via a non-square weighting matrix

Let $W$ be a full-rank $m \times n$ matrix with $n<m$, i.e. it has linearly independent columns Define the wieghted norm on $\mathbb R^n$ as $\|x\|_W=\|Wx\|_{\infty}$. Is there a formula for the ...