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Questions tagged [linear-algebra]

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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If $\langle T v,v\rangle=\langle v,v\rangle$ for all $v\in V$ satisfying $\langle v,v\rangle=1$, then $T$ is the identity.

I'm trying to prove the following: Let $V$ be a vector space over $\mathbb{C}$ and let $T$ be a linear operator on $V$. If $\langle T v,v\rangle=\langle v,v\rangle$ for all $v\in V$ satisfying $\...
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Polynomial Multiplication and Division by Monomial

Multiply: (9x5+5x2−6x)(5x6−2x3−7x)(9x5+5x2-6x)(5x6-2x3-7x) = ?
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1answer
11 views

Intersecting Angle between 2 Line Segments

I asked a similar question here before, which I'll link to here. I realized, however, that I needed to extend the question from the angle between a point and a line segment to the angle between 2 line ...
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2answers
13 views

For which $x\in\mathbb{R}$ does the vector system rank third

Someone can help me to define the rule of this task? For which $x\in\mathbb{R}$ does the vector system rank third $\{\begin{bmatrix}x\\1\\3\end{bmatrix}$,$\begin{bmatrix}1\\3\\-2x\end{bmatrix}$,$\...
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2answers
25 views

Determine the linear mapping $g$ belonging to matrix $B$

Determine the linear mapping $g$ belonging to matrix $B$, where $$ B = \begin{bmatrix}1 & 0\\1 & -1\end{bmatrix}. $$ I don't understand this task. How could I find the linear map of a matrix?
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1answer
37 views

Hyperplane in the first dimension?

If a hyperplane in $\mathbb{R}^n$ is of dimension n-1, what is the geometric interpretation of a hyperplane in $\mathbb{R}^1$? Is it a point?
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0answers
15 views

Searching for matrices with linear dependent rows with some property

I am looking for matrices $X \in \mathbb{R}^{n \times m}$ depending on some variable $x$ with linearly dependent rows such that there does not exists some $y \in \mathbb{R}^m$ with $yX(x) > 0$ (...
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2answers
25 views

If f is a linear transformation on a finite dimensional vector space V…

If f is a linear transformation on a finite dimensional vector space V satisfying $f^2=f$, explain how to find a diagonal matrix representing f This is a workbook question im lost on, could someone ...
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1answer
20 views

Definition of Clifford Algebra

Cliffard algebra defined by relation: $x*y+y*x=g(x,y)1$, where g(x,y) is bilinear symmetric form. What does mean $g(x,y)1$, why it's not just $g(x,y)$, without the identity?
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2answers
15 views

Finding Vector in New Coordinate System

Let $B = \{(1,2),(2,3)\}$ and $C = \{(-1,1),(2,1)\}$ be bases for $\Bbb{R}^2$. If $[v]_{C} = (1,2)$ what is $[v]_{B}$? The problem doesn't specify what $v$ is, so I'm assuming it is some abstract ...
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1answer
50 views

Meaning of an $a^T$

-- Edited -- For $a \in \mathbb{R}^n$, $a^T a$ means its norm. What does mean $a a^T$? Best regards, Bruno
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show that $(x_n)_{n\geq0}$ and $(y_n)_{n\geq0}$ are l.i. in $Fib$ => $(x_0,x_1);(y_0,y_1)$ are l.i. in $\mathbb{R}^2$

Let $Fib=\{(x_n)_{n\geq0}|x_{n+2}=x_{n+1}+x_n, x_n\in\mathbb{R}\}$. Determine a base in Fib. My attempt: It is easily to be seen that $Fib$ is a vectorial space over $(\mathbb{R}, \oplus,\odot)$. ...
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0answers
19 views

Relation between Laplace solution of differential equations and the Pseudo-inverse

I am trying to understand the relation between the solution of differential equation in Laplace space and matrix inverse/pseudo-inverse problems. Consider the system of ODEs: $$\dot{\mathbf{x}}(t) = ...
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0answers
29 views

Find base in a $4\times 4$ matrix.

Determine a base in the matrix $A$ and determine dimension of $\ker(A)$ and dimension of $\operatorname{Im}(A)$. With $$A=\begin{pmatrix}1&-2&4&-1\\3&-1&2&-1\\-1&-3&...
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0answers
24 views

about tensor product

I understand that bilinear map only problem is why $\beta_s$ ($1 \otimes x_i$, $1 \otimes x_j$, ) is invertible? Also how to do this argument for projective case?
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2answers
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Computation of a projection onto orthogonal complement

In $\mathbb{R}^4$, let $\underline{a} = (1, 2,-2, 0)$ , $\underline{b} = (2, 1, 0, 4)$ and $\underline{c} = (5, 7, 3, 2)$. I want to determine that the orthogonal projection of $\underline{c}$ on $&...
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1answer
33 views

Dimension of kernel of a linear map $\phi: M_n(\mathbb R) \to \mathbb R^n$

Let $A \in M_n(\mathbb R)$ be a fixed matrix. Let us define a linear map $\phi: M_n(\mathbb R) \to \mathbb R^n$ by \begin{align*} X \mapsto (XA-AX)e_1, \end{align*} where $e_1$ is the standard basis ...
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1answer
27 views

Are there terminologies for ($A A^T$ or $A A^H$) and ($A^T A$ or $A^H A$)?

Are there terminologies for $A A^T$ and $A^T A$, respectively, where $A$ is a matrix? Like "$A A^T$ is the (something) of $A$." I know that if $A$ were a vector, we could use the terms inner product ...
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0answers
16 views

Decompose an equation into a matrix form

How can I decompose the following equation into a set of matrix multiplications? $f(x',\alpha,\beta) = \sum_{n=0}^{\infty}\sum_{n'=0}^{\infty}\sum_{l=0}^{n}\sum_{m=-l}^{l}A_{nn'l}(e^{(n-l)x'}-e^{(n'-...
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1answer
21 views

$A\in M_{m\times n}(k)$ is one one iff row rank is $n$

$A\in M_{m\times n}(k)$ is one one iff row rank is $n$. Here $k$ is a field. My Attempt 1): $A$ is $1-1$ $\iff$ker $A$ = $\{0_{m\times1}\}$ $\iff$All cols are linearly independant $\iff$Col Rank is $...
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0answers
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Does it make sense to write $(T - \lambda I)v$ if $T$ is a linear transformation?

I am reading this page to try to understand Jordan form. $V$ is a finite-dimensional complex vector space, and until now $T$ has always represented an "operator", by which I guess they mean a linear ...
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Is A a subspace of P3

A = {(a0) +(a1)x+(a3)x^3 : a0,a1,a3 ∈ R} I've looked through similar examples, but they either all have conditions like 'a0>a1' or don't explain what to do with the x variables. I think i have to ...
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2answers
31 views

Showing a basis of $V$ over $\mathbb C$

Given that $V$ is over $\mathbb{C}$ and $\{u, v, w\}$ are distinct and a basis of $V$. Show that $$\{u - (1 +i)v, u+v+w, -2iu\}$$ is also a basis. So far, I have just started by stating that they ...
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0answers
16 views

Online tool for practicing matrix operations

I've decided to brush up on Linear Algebra after several years of disuse, and have rediscovered how tedious it can be to write out matrices by hand. Is there some online tool I can use to speed the ...
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0answers
11 views

Rearranging the multivariate Gaussian distributions into squared form

If we have a multiplication of two multivariate Gaussian distributions, we can use sum of the squared form given in section 8.1.7 of this book. Now, I have a situation as follows: \begin{equation} -\...
2
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2answers
27 views

Least squares solutions of matrices with redundant columns?

There was a similar question here, but I either did not understand the answers or the answers were too general. I am wondering specifically how to find the solutions. For example, what are the least ...
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0answers
14 views

How do fractional tensor products work?

In this blog post, Terry Tao discusses the $n$-fold tensor product of a one-dimensional vector space $V^L$ ($L$ is just a non-numeric label, not an exponent). He claims that With a bit of ...
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1answer
25 views

If $A$ is positive semidefinite, then $A=USU^T$?

If A is a positive semidefinite matrix, then it has a singular value decomposition $A=USV^T$ with $U=V$. My textbook states this as fact, but I cannot seem to prove it. Additionally, $S$ must have the ...
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2answers
42 views

Is there an elegant way to determine $Av$ given $Au_1, Au_2$, and $Au_3$ for a $3\times3$ matrix $A$?

Let A be a 3x3 matrix such that ${A} \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ 7 \\ -13 \end{pmatrix}, \quad \ {A} \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} = \begin{pmatrix} -6 ...
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1answer
16 views

Solving system of overdetermined linear equations with $2$-norm

Given a system of overdetermined linear equations $Ax = b$, I know that it is possible to find a solution of that problem using $2$-norm, but I have a doubt: Is it wrong to think that it is equivalent ...
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2answers
25 views

Idempotent linear transformation from $V$ to $V$ is the direct sum of $\operatorname{range}(T)$ and $\operatorname{null}(T)$

The problem statement is: If $T\in\mathcal{L}(V)$ and $T^2=T$ (idempotent), prove that $V=\operatorname{range}(T)\oplus \operatorname{null}(T)$. I'm not exactly sure where to start with this ...
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1answer
24 views

Finding the dual of a Linear program

suppose we have $$ min z = 6 x_1 + 20 x+3 $$ such that $$ x_2 + 4 x_3 \leq 10 $$ $$ - x_2 +2 x_3 \leq 11 $$ $$ x_i \geq 0 $$ Find the dual. ATTEMPT: I wrote $$ max = 10 y_1 + 11 y_2 $$ st ...
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1answer
13 views

Writing a vector as a linear combination of vectors from another basis

I have the bases $B=\{\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \end{pmatrix}\}$ and $C=\{\begin{pmatrix} -4 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{pmatrix}\}$. I'm asked ...
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0answers
18 views

Multiplication of two matrices using convolution

Let's say I define my matrix $M$ such that $$m_{i,j}=m_{i-1,i-j}$$ Now, that means we only need the first column and row of this matrix as all the other elements in that matrix are just repetitions of ...
1
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1answer
18 views

Operator norm of a family of matrices

Let $c$ be a complex number. Consider the family of $n\times n$ matrices $M_n$ which have $c$'s on one off-diagonal, $\bar{c}$'s on the other off-diagonal, and zero everywhere else. So $M_4$ looks ...
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0answers
16 views

Function in definition of linear differential equation

The first chapter of an introductory book on differential equations (Boyce Diprima) has the following definition of linear differential equation: The differential equation $F(t,y,y',..., y^{(n)})=0$ ...
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3answers
49 views

If $A=PDP^T$, does $P$ have to be orthogonal.

A matrix A is called orthogonally diagonalizable if $A=PDP^{-1}$ and $A=PDP^{T}$, where $D$ is diagonal. Therefore, $P^{-1}=P^T$ and thus $P$ is an orthogonal matrix. If you are only given the fact ...
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2answers
41 views

Four Dimensional intersection point

How can I find the intersection point of two linearly independent 4D dimensional planes? I know that the first plane A goes through the four-dimensional points A1(1,2,3,4) A2(0,1,0,1) A3(0,1,1,0) and ...
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1answer
30 views

About eigenvalues, matrix equation with non invertible matrices

Let A,B be two non-invertible square matrices of the same size. Would there be a general procedure for solving the equation $$ Ax = Bkx $$ for both $k$ and $x$? ($k$ is a scalar). For example, if ...
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0answers
23 views

Get the determinant of a block matrix given the submatrices.

I have a matrix $M$ that is equal to: $\begin{bmatrix} 1 & 1 & 0\\ 1 & 0 & 1 \\ 0 & 1 & 1\\ \end{bmatrix}$ It's easy to compute $|M| = -2$, but then given matrix: $ N = \...
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2answers
15 views

Matrix Representation of Linear Transformation from R2x2 to R3

We have a linear transformation T : $\mathbb R^{2\times2} \rightarrow \mathbb R^{3}$ defined by $$T\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)=(a+b,2d,b).$$ Let A and B be the ordered ...
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1answer
16 views

Question in the lecture of Lineare algebra done right, exemple of subspace

Can sombothy explain why the set of differentiable real-valued functions $f$ on the interval $(0,3)$ such that $f'(2)= b$ is a subspace of $R^{(0,3)}$ if and only if $b = 0$. Thanks
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1answer
20 views

Writing cycles of a graph as a linear combination of fundamental cycles

It is folklore that the fundamental cycles (corresponding to aparticular spanning tree) of a graph constitute a basis for its cycle space, while the proof uses the linear indepence of fundamental ...
3
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1answer
23 views

Is it true that: $(U + W)\cap S = U \cap S + W\cap S$?, where $U,W,S \subseteq V$ where $V$ is a vectorial space.

Is it true that $(U + W)\cap S = U \cap S + W\cap S$?, where $U,W,S$ $\subseteq V$ where $V$ is a vectorial space. My attempt: $$\begin{array}{l}U=\{\lambda(\begin{array}{c}u_1\\\vdots\\u_n\end{...
4
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1answer
28 views

Question related to the trace and the commutator of matrices

Let $K$ be any field and $n\in \mathbb N$. For every $A\in M_n(K)$, define a linear form $\lambda_A: M_n(K) \rightarrow K$ by sending the matrix $M$ to $\lambda_A(M):= \operatorname{Tr}(AM)$. The map $...
2
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1answer
39 views

determinant of a complex matrix

Suppose we consider a complex $n\times n$ matrix $A$. It can be writen generally as $A=U+iV$, where $U$, $V$ are $n\times n$ real matrices. Now I hope to study the determinant of the following matrix ...
4
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1answer
19 views

If $M$ is symmetric posdef, then all diagonal band matrices derived from $M$ are also posdef?

Let $M$ be a positive definite and symmetric matrix: $$M = \left(\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n}\\ a_{21} & a_{22} & \cdots & a_{2 n}\\ \vdots &...
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0answers
13 views

Is it correct to perform operations on matrices inside of matrices rather than the whole matrix?

It seems to be a generally useful technique when proving theorems about matrices to consider smaller matrices inside of other matrices. For example, I might write $$M = \begin{pmatrix} I_{n \times n}...
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0answers
13 views

given a system of N linear equations, Is there an algorithm that can find a solution that solves the most number of equations in this system

My apologies if this question makes no sense; I am trying to find an algorithm that can solve a linear system of equations. Unlike most problems like this- for this particular case, this algorithm ...
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0answers
8 views

Why every linear independent system of vectors in a subspace is also linear independent in the respective vectorspace?

I cite my German book: [e means element of] [dim-f is the dimension under the field f] [c means subset of] corollary 12: If V is a f-vectorspace, f is a field and n e N. Then is equivalent: (i) ...