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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1answer
18 views

Find the value of $a$ for which two lines are perpendicular

Two vectors are perpendicular when their dot product is zero. But how to get the dot product of two lines? $r_1= (8,-3,1)+s(12,-5,0)$ $r_2= (1,14,3)+t(5,a,b)$ The question asks for the value of ...
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1answer
53 views

Why must $b=0$ for this linear system to have infinitely many solutions for all $a$?

Consider the parameterized linear system of equations represented by the augmented matrix: $$ \left[ \begin{array}{ccc|c} 1 & 0 & a & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & ...
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1answer
30 views

Proving result on matrix rank

Is it true that, if $A=QR$ with $Q$ unitary matrix and $R$ an upper triangular matrix, and $A\in\mathbb{C}^{n\times n}$, then the rank of $A$ is the same as that of $R$? And if so, how do I prove it?
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3answers
55 views

Index notation for inverse matrices

I have a question: There is an standard way to write the inverse of a matrix in index notation?. The reason is that I don't want to write $(A^{-1})_{ij}$ or $(A^{-1})_i^j$ or $(A^{-1})^{ij}$ using ...
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1answer
39 views

Determine Whether The Equation Is Linear In $x_1, x_2, $ And $x_3$

Determine Whether The Equation Is Linear In $x_1, x_2, $ And $x_3$ According to the book I'm using, the following are linear equations: $$x_1 + 5x_2 - \sqrt{2x_3} = 1$$ $$ \pi x_1 - \sqrt{2x_2} + ...
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2answers
56 views

$\dim B/A=\dim B-\dim A$?

If $A,B$ are two vector spaces over $k$ such that $B\subseteq A$, can I say $\dim B/A=\dim B-\dim A$? I need of this result to prove a theorem I'm working on. Thanks in advance
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2answers
88 views

Minimum linear subspaces cover problem

Given a set of vectors $V=\{v_1,v_2,...,v_n\}$ and $m$ vector sets $V_1,V_2,...,V_m$ ($V_i$ may not be a subset of $V$), I want to find minimum number of sets from $\{V_1,V_2,...,V_m\}$, denoted as ...
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1answer
43 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
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0answers
17 views

Verification of a condition when sum of two normal operator is also a normal operator

Let $A$ and $B$ be two normal operator on a inner product space $V$ and $AB=BA$. Is it true that $A+B$ is also a normal operator on $V$? Can we also say that $AB^\dagger=B^\dagger A$ and $A^\dagger ...
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1answer
68 views

Self-adjoint on dot product

Let be $V = M_3(\mathbb{R})$ the vector space of the real antisymmetric matrix and let be $\phi$ the scalar product defined by $\phi(X,Y) = tr(^tXY)~ \forall X, Y \in V$. Let be $A$ a symmetric ...
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1answer
24 views

Finding eigenvectors for the largest eigenvalue vs one with the largest absolute value

If I want to solve a generalized eigenvalue problem such as: $$A x = \lambda x$$ The problem is to find eigenvectors corresponding to the largest eigenvalues (sometimes in an optimization problem ...
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1answer
16 views

Does the map $F(x, y)=(f(x, y), y)$ induce an isomorphism $dF_{(a, b)}$?

Suppose $f:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ be a $C^p$ map such that $df_{(a, b)}:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ is surjective for some $(a, b)\in\mathbb R^{n+m}$. Define ...
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2answers
112 views
+50

Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all ...
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2answers
28 views

How do you find the change of coordinates matrix from a given matrix to the standard basis?

I'm not sure how to approach this problem. The examples I've come across on the internet show how to find the change of coordinates matrix from a matrix to another matrix, such as B to C (for ...
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2answers
41 views

An equation with square root

$$2x+\sqrt{1-3x}=0$$ I know thats a basic task, but I forgot how to do such simple equations. Should I just switch 2x to the other side, and square both sides?
3
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1answer
34 views

Integration by parts for Matrices

I understand how to do integration by parts for individual functions. I am trying to apply integration by parts to matrices/vectors where the order of terms is important. So say I have a matrix A ...
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1answer
35 views

Is the map $T(u, v)=(A(u, v), v)$ surjective?

Let $A:\mathbb R^{m+n}\longrightarrow \mathbb R^n$ be a linear surjective map and let $T:\mathbb R^{n+m}\longrightarrow \mathbb R^n\times \mathbb R^m$ be the linear map given by $$T(u, v)=(A(u, v), ...
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1answer
29 views

Eigenvalues of ad (Adjoint action) in semisimple lie algebra?

Suppose $V=V_0\oplus V_1$ be a $Z_2$-graded semi-simple lie algebra and, $\xi\in V_1$. The maps $ad_\xi \circ ad_\xi :V_0\longrightarrow V_0$ and $ad_\xi \circ ad_\xi :V_1\longrightarrow V_1$ are ...
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0answers
33 views

Solving system of equations in rationals

Do there exist solutions to solve system of $n-2$ equations with $n-2$ variables where $n$ is fixed even integer and $a_i,b,c\in\mathbb{Q},i\in\{0,1,2,\cdots,n-5\}$ $$\left\{ ...
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1answer
26 views

Transpose of higher dimension matrices

We all know transpose of 2D matrix A Old $A_{ij}$ will be replaced by $A_{ji}$ in the transpose matrix and vice versa Question If A is 3D matrices of $3\times 3 \times 8$ then what is old ...
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2answers
29 views

$Ker(T) \subseteq V$ Is A Subspace

Let $V,W$ be a vector space over a field $\mathbb F$, and $T$ a linear transformation $T:V \rightarrow W$ $Ker(T) \subseteq V $ to prove that $Ker(T)$ is a subspace can we say that: by definition ...
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1answer
33 views

Diagonalization of a matrix with change of basis

I was trying to diagonalize a not really nice matrix doing first a change of basis but I noticed that the two characteristic polynomials I get are different. Original matrix and its characteristic ...
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1answer
40 views

Isomorphism between $E$ and $Lin(E)$ : infinite dimensional case.

Does $E$ and $Lin(E)$ where $Lin(E)=\{A : E \rightarrow E ∣ A \quad\text{is linear}\}$ are isomorph if $E$ infinite dimensional case ? I know that if $E$ is finite dimension the result is true ...
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2answers
22 views

Mean line from several lines

i want to find the average line from different line segments. I don't know how to do this and also don't know how to descripe it specifically. So I'll give you an example: If a have two lines with the ...
2
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0answers
17 views

Counting the operations of a problem

I have a square matrix $A\in\mathbb{R}^{n\times n}$, it has a LU decomposition. $L$ and $U$ are triangular and $L$ has ones on the main diagonal. I'm counting the number of operations for ...
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2answers
25 views

Linear Transformation $T_{A}$ Is invertible $\iff$ A Is invertible

Let $T_{A}$ be the linear mapping corresponding to the matrix A, and $A \in F^{n*n}$ $T_{A}$ Is invertible $\iff$ there is $T_{A}^{-1}$ so $T_{A} \circ T_{A}^{-1}=I $ $T_{A} \circ T_{A}^{-1}(v)=v$ ...
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0answers
28 views

Orbits of $Sp(n,R)$ under action of $Gl(2n,R)$ by conjugation

These questions arose from a question related to K-theory, I am hoping for (big) results from the theory of linear algebraic groups to be helpful. Maybe somebody with a better background there can ...
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1answer
65 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
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1answer
16 views

A hyperplane in a $k$-algebra

Let there exist a nonsingular bilinear pairing $B:R×R→k$, where $R$ is a finite dimensional algebra over a field $k$, such that $B(xy,z)=B(x,yz)$ for all $x,y,z$ in $R$. Why the set $\{z∈R∶B(1,z)=0\}$ ...
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1answer
22 views

Basic question about similar matrices

Let $A, B$ be similar $n \times n$ matrices over (say) $\mathbb{R}.$ Is it the case that $A+cI$ is similar to $B+cI$? Here is an argument (which seems too good to be true...) Since $A, B$ are ...
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2answers
37 views

Proving that only the linear codes pass parity check

An exercise in my book goes as follows: Let $C$ be a binary $(n,k)$ linear code with parity-check matrix $H$. We know $Hc=0$ for all $c\in C$. Show that $Hw=0$ implies $w\in C$. My idea: Let ...
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1answer
47 views

Tensor product of Frobenius algebras

In proving the fact that the tensor product of any two finite-dimensional Frobenius algebras $R$ and $S$ over the same field $k$, it is usually defined a $k$-bilinear pairing $E: W×W→k$ where ...
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1answer
26 views

Quadratic form in canonical form relation [closed]

The homogeneous quadratic form can be written as a matrix. It is also written as a canonical form by using orthogonal transformation. Why we are going for canonical form and what is the relation ...
2
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1answer
45 views

Calculating Vandermonde determinant

I understand that the Vandermonde determinant $$ W(x_1, \ldots, x_n) = \left| \begin{array}{cccc} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & ...
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1answer
22 views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 ...
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1answer
53 views

When and why can functions “take on” the role of vectors in defining vector speaces?

In what I call "advanced" linear algebra, we examine the properties of vectors in a vector space like an inner product space by checking that they satisfy e.g. the Cauchy-Schwartz inequality, the ...
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1answer
24 views

Describe the solution set of the system

Consider the linear system below: $$\begin{array}{ccccccc} x_1&-&2x_2&+&&&x_4&=&1\\ 2x_1& -& 5x_2& -& 2x_3& +& k^2x_4 &= &-2\\ ...
4
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2answers
196 views

Prove that $A \circ B = AB$ if and only if both $A$ and $B$ are diagonal

Definition. Hadamard product. Let $A,B \in \mathbb{C}^{m \times n}$. The Hadamard product of $A$ and $B$ is defined by $[A \circ B]_{ij} = [A]_{ij}[B]_{ij}$ for all $i = 1, \dots, m$, $j = 1, \dots, ...
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4answers
64 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
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1answer
31 views

Matrix Multiplication - When do you only multiply by one number and add vs. multiplying all numbers?

*I wasn't sure where to put this. Just let me know if I should delete it or if there is another category/website where this question would fit better. Thanks! Or if you know the answer & don't ...
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2answers
104 views

Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [on hold]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
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1answer
27 views

Proving $\mathrm{Hom}(V \rightarrow W)$ is a vector space

It can easily be proven that $\newcommand{\Hom}{\mathrm{Hom}}\Hom(V \rightarrow W)$ is a sub-space. 1. we know that for any $T:V\rightarrow W$, T(0)=0, therefore $0\in \Hom(V \rightarrow W)$ 2. ...
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2answers
44 views

Linear Code (9,5): Is my Parity Check correct?

I have an exercise about designing parity checks for the Hamming (9,5) group code with minimum distance $3$. I use the following notation for the generator matrix: $$ ...
1
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1answer
57 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
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2answers
29 views

problem with denominator in transformation

hi i cant understand where the 2 comes from in this transformation any help would be appreciateD
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1answer
40 views

calculus / algebra

Hi can anyone go through the transformation of the equation below as i cannot understand where the 2 in comes from any help would be much appreciated $$\frac{\omega k^{0.5}}{\omega k} = ...
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0answers
22 views

Positive definite [closed]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
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1answer
38 views

Determinant of a rank-one update of a scalar matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Eigenvalues of a matrix of $1$'s ...
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1answer
58 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
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1answer
36 views

Positive definite matrix. [closed]

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?