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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Express in terms of the spectral decomposition of $A$ the set of $x, y$ for which an inequality is satisfied

I'm confused by this problem: Let $A \in \mathbb{C}^{n \times n}$ be diagonalizable with eigenvalues $0 \leq \lambda_1 \leq \cdots \leq \lambda_n$. Express in terms of the spectral decomposition ...
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1answer
62 views

If A is 5 by 3 and B is 3 by 5 (with dependent columns), Is $AB = I$ impossible?

Let me first introduce the problem. This is part of the quiz problem from MIT's 18.06 course (Spring 2012 semester, quiz 1, problem 3). My question is related to (b) but (a) is mentioned in the ...
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11 views

Solutions to Binary Equations

Let $A\in M(m,n,\{0,1\}$) (i.e. $m \times n$ matrices with entries in $\{0,1\}$) and $x,y\in \{0,1\}^n$. We will denote the $i$-th row of $A$ as $row_i(A)\in \{0,1\}^n$. Define, $ z_i = \begin{...
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1answer
23 views

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$.. [on hold]

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$ dimensional subspace of $\mathbb{R}^n$ that's contained in $H$, show that $G = H$ I know that a subspace of $\mathbb{R}^n$ is ...
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1answer
19 views

Interchanging vectors coordinates

Is there any relation between two vectors with interchanging coordinates .. i.e: the x component of the first is the y component of the second and vice versa.
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35 views

Answer checking - involved derivative under summation

I'd like someone smarter and more experienced than me to check my answer and give advice on how to do it better and derive a closed form for what I'm looking for. Given a matrix $Y \in \mathbb R^{m \...
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2answers
42 views

The first 'primordial' basis of a finite vector space

Let's take a vector space $V $ and set $V= \mathbb{R}^3 $ for ease of mind. Usually we equip $V $ with the standard basis $\{e_1,e_2,e_3\} $ and we express all our vectors in that basis: $$v = (a,b,...
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4answers
42 views

What is the calculation behind this linear algebra basic question?

$$ \begin{bmatrix}1&2\\3&8\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}\ \text{yields}\ \begin{bmatrix}1&2\\0&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\...
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0answers
14 views

Is there a correct way to compute matrix K, for SVD, and eigenvalue analysis

I'm reading two different papers on multivariate data analysis, both use the matrix $K$ for singular value decomposition, eigenvalues and eigenvectors. This is the case where there are two matrices, ...
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1answer
26 views

Reducing row of a matrix with 1 parameter

I have this matrix: $$\begin{bmatrix}a-2 & 2 & 3 & 0 & -1\\ 0 & 1 & 1 & -1 & 0\\ 0 & 0 & 2(1+a) & 1+a & 3\end{bmatrix}$$ I need to find what values ...
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1answer
23 views

$K$-algebra homomorphism

Let $\mathbb{T}_n(K)$ the set of all triangular matrices, where $K$ is a field. And let $U$ the set of all matrices $\lambda=[\lambda_{ij}]$ in $\mathbb{T}_n(K)$ with $\lambda_{ii}=0$. I want to ...
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0answers
26 views

Solving Systems of linear equations between a square matrix and a rectangular matrix with block decomposition

I am trying to decompose solving a system of linear equations using block decomposition where I have an (n x n) matrix A, which is a lower/upper triangular matrix, and a matrix B, which is a ...
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32 views

Generate function from data

I have a series of inputs and outputs : Inputs -> Outputs 1,2,3 -> 4 4,5,6 -> 5 7,8,9 -> 6 Is there a field of study that can generate a single ...
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0answers
14 views

How to linearize two discrete maps with time delay feedback

I have a 2-D system of two discrete maps $x_{n+1} = f(x_n) + P_1(y_n - y_{n-1})$, $y_{n+1} = g(y_n) + P_2(x_n - x_{n-1})$ with $g,f$ being smooth functions and $P_1, P_2$ belonging to the reals ...
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0answers
29 views

prove that $A^{-1} = (1/detA) \operatorname{cof} A^T$ [on hold]

Can you please explain to me how to prove this theorem? Theorem: if $\det(A)\ne 0$, then $A$ is invertible and $A^{-1} = \frac 1{\det(A)} \operatorname{adj} A$
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1answer
28 views

About the eigenvalues of a block Toeplitz (tridiagonal) matrix

I have found the following $n\times n$ squared matrix in one stability analysis problem (i.e. I have to identify the sign of its eigenvalues) $$ A(\theta) = \begin{bmatrix} W(\theta)+W(\theta)^T & ...
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1answer
18 views

How to prove or disprove there is a unique solution to this linear system where the variable are constrained to the interval [0,1].

Be $m$ and and $n$ integers such that $1\leq m \leq n$. Be $l_k$ variable with $dom(l_k)=[0,1]$. Take the folowwing linear equation system: $\sum_\limits{k=1}^{n} l_k =1$ $\sum_\limits{k=1}^{n} k\...
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1answer
40 views

Show that the trace of A is less than n

Let $A$ be an $n\times n$ matrix with complex entries such that $A^k=I_n$ for some positive integer $k$. Show that the trace of $A$ satisfies $$|tr(A)| \leq n.$$ I have no idea how to approach this ...
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2answers
26 views

The Matrix of a reflection (around abitrary plane)

Let $\Upsilon :\mathbb{R}^3\rightarrow \mathbb{R}^3$ be a reflection across the plane: $\pi : -x + y + 2z = 0 $. Find the matrix of this linear transformation using the standard basis vectors and the ...
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0answers
11 views

when the k-algebra product is a vector [on hold]

I have a question, maybe stupid because I'm physics student and so I'm not following algebra, tensor calculus courses. I hope to clarify well. as it was written on page 323 in this book a product on ...
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1answer
23 views

Theorem regarding direct sums

Let $w_1$ and $w_2$ be subspaces of V. Prove that V is direct sum of $w_1$ an $w_2$ iff each vector in V can be uniquely written as $x_1 + x_2$ where $x_1$ belongs to $w_1$ and $x_2$ belongs to $w_2$ ...
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1answer
18 views

Construction of a Module isomorphism

Let $R$ be a PID. Consider the sets $X_0=\{v_0,v_1,v_2\}$ and $X_1=\{e_1,e_2,e_3\}$ and let $C_i$ be the free $R$-module on $X_i$ for $i=0,1$. Consider the $R$-module homomorphism $$C_1\;\...
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1answer
29 views

Overlap between two vectors

Given are two vectors ${\bf g}_1, {\bf g}_2\in\mathbb{R}^N$ with non-zero scalar-product ${\bf g}_1^\top{\bf g}_2 \ne 0$. Then there exist three unique orthogonal unit vectors ${\bf e}_1, {\bf e}_2, {\...
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0answers
31 views

How to randomly generate two integer matrices $A$ and $B$, so that entries of 3 metrics $A$, $B$, and $AB$ are within certain range?

I ran into this question when writing a program. I need to generate two matrices, and calculate their product. However, I must ensure all entries are within 8-bit signed integer range, i.e. $[-128, ...
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16 views

Water drop evaporation time and contact angle

I'm measuring water drop evaporation on different surfaces and it would be nice to have an equation to roughly estimate evaporation time (or contact angle). Some drops are hydrophobic, others ...
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0answers
36 views

What is a linear isomorphism?

I am working with the book Manifolds and Differential Geometry from Lee and I am a little bit puzzled since he sometimes talks about linear isomorphism (proposition 2.3 for example). But isn't an ...
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0answers
37 views

Finding a linear system to solve quadratic equations

considering an equality with a polynomial of second degree where the coefficient for $x^2$ is $1$ I know that $$ a x^2 + b x + c = a(x-\alpha)(x-\beta) = 0 $$ I also know that $$ \alpha + \beta = -...
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2answers
76 views

How come two of the eigenvalues are same?

Question is about finding the eigenvalues of the matrix : $$\begin{bmatrix} 0 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ \end{bmatrix}$$ the matrix would become $$\begin{bmatrix} -...
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1answer
33 views

Least Squares Algorithm with Inverse Norm

Given an overdetermined linear system $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m \times 1}$ with $A < 0$ and $b < 0$. What is a good way to numerically determine $$ \min_x \left\lVert ...
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0answers
36 views

Real and imaginary part of tensors of matrices

Given a matrix $A\in \mathbb{C}^{n\times m}$, clearly we can write $A=\Re(A)+i \Im(A)$, i.e., the real and imaginary part of $A$. (For instance, $A=[1,i]$, then $A=[1,0]+i[0,1]$). I am interested in ...
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39 views

what is Expected Mean

Thus the expected mean $\mu$ of the set $\mathcal S$ can be given as \begin{align*} \mathbb E \mu&= \sigma^2+\frac 1r \sum_{i=1}^m\left(\mathbb E\lambda_i-\sigma^2\right)\\ &\geq \sigma^2+\...
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1answer
19 views

Calculating the missing two points of rectangle if 2 points and the aspect ratio are known

How can I calculate the missing two points of a rectangle if I know 2 points (top left and top right) and the aspect ratio i.e 16:10. For example: Top left: A(834, 449) and Top right: B(1675, 423)
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1answer
15 views

Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
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1answer
30 views

Question on proof of number of solutions of linear system

The proof my book uses starts off by saying: "If the system has exactly one solution or no solutions, then there is nothing to prove", and then continues on by assuming there is an infinite ...
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0answers
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Curious about the deduction procedure on shannon transform and stieltjes transform

I found some scholars clarified that the relationship between the Shannon transform and Stieltjes transform with \begin{equation} \frac{\gamma}{\log e}\frac{d}{d\gamma}\mathcal{V}_N(\gamma) = 1 - \...
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1answer
40 views

prove subspace of a vector space

Consider the subset $T$ of $\mathbb{R}^2$ defined as follows: $T := \left\{(x, y) : x, y \in \mathbb{R} : y = 3x \right\}$. Prove that T is a subspace of the vector space $\mathbb{R}^2$. My attempt: ...
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3answers
75 views

Which non-negative matrices have negative eigenvalues?

It's easy to proof by counterexample that non-negative matrices can have negative eigenvalues. For example, the following matrix have -1 as an eigenvalue: $$ A = \begin{bmatrix} 0 & 0 & 0 ...
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0answers
43 views

Polar coordinate in Cartesian

My book states one can write polar coordinates $(\hat{r}, \hat{\theta})$as $$\hat{r} = \cos \theta i + \sin \theta j$$ $$\hat{\theta} = -\sin \theta i + \cos\theta j$$ Can someone explain how $$\hat{...
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1answer
20 views

Is there a unique projection map in this case?

Let $X$ be a Banach space over $\mathbb{C}$. Let $A,B$ be closed subspaces of $X$ such that $X=A\oplus B$. Assume that $||a+b||=||a||+||b||$ for each $(a,b)\in A\times B$. Then, does there exist a ...
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2answers
41 views

Finding an orthonormal basis for the plane $x_1 - 5x_2 - x_3 = 0$

Find an orthonormal basis of the plane $x_1 - 5x_2 - x_3 = 0$ I'm having trouble with this problem. So I picked the vectors $u_1 = \begin{bmatrix}1\\0\\1\end{bmatrix}$ and $u_2 = \begin{bmatrix}5\\...
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1answer
43 views

How to project $x_2$ onto $u_1$

I'm following a solution from here (the first problem), I don't understand how to "project $x_2$ onto $u_1$" 1) how does:$\begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix}$ ...
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1answer
18 views

Choosing the seed for a LFSR

I was just wondering how the seed of a LFSR is chosen and is there any connection between the seed chosen and cryptographic strength of the keystream generated? Thank you
3
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3answers
51 views

If $T$ and $T^2$ have equal rank then $V=\ker T\oplus {\rm im}\, T$ for $V$ finite dimensional.

I am trying to prove the following: Let $V$ be a finite-dimensional vector space. Consider an operator $T$ on $V$ such that $\text{dim range}(T)=\text{dim range}(T^2)$. Show that $V=\text{null}(T)\...
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1answer
29 views

How to rotate a coordinate system in $\mathbb{R}^3$ through an angle about an arbitrary axis passing through origin?

The question spurred in my mind when I was asked the following: Find the transformation matrix T that describes a rotation by $120^\circ$ about an axis from the origin through the point $(1,1,1)$....
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1answer
35 views

Does the line pass through the origin? [on hold]

Does the line $L: (3,4,-1)+s(7,-2,1),\ s\in \Bbb R$ pass through the origin? I'm not sure how to do this.
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25 views

Conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z}) × GL_m(\mathbb{Z}/q\mathbb{Z})$ [on hold]

Find the number of conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z})× GL_m(\mathbb{Z}/q\mathbb{Z})$ of cyclic subgroups of order pq?.
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2answers
36 views

Determine the distance between the point $(2,-3,1)$ and the point of intersection of three planes

Determine the distance between the point $(2,-3,1)$ and the point of intersection of the following system. $3x-y+z=4$ $-x+2y+3z=7$ $x+3y+4z=12$ I'm not quite sure how to do this?
3
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2answers
22 views

(Double) Coset of $GL(n, q^2)/GL(n, q)$

I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the ...
2
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0answers
29 views

Geometry Of Unitary Transformations

Ever since I first took Linear Algebra, I have over time realized how concepts like determinants, eigenvalues, diagonalization, orthogonal transformations and so on have very intuitive geometric ...