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 ...
77,387 questions
-1
votes
0answers
10 views
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 $\...
-7
votes
0answers
14 views
Polynomial Multiplication and Division by Monomial
Multiply: (9x5+5x2−6x)(5x6−2x3−7x)(9x5+5x2-6x)(5x6-2x3-7x) = ?
1
vote
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 ...
0
votes
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
$\{\begin{bmatrix}x\\1\\3\end{bmatrix}$,$\begin{bmatrix}1\\3\\-2x\end{bmatrix}$,$\...
1
vote
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?
1
vote
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?
1
vote
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 $yX(x) > 0$ (...
0
votes
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 ...
1
vote
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?
1
vote
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 ...
2
votes
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
0
votes
0answers
18 views
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 $(\mathbb{R}, \oplus,\odot)$.
...
2
votes
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:
$$\dot{\mathbf{x}}(t) = ...
0
votes
0answers
29 views
Find base in a $4\times 4$ matrix.
Determine a base in the matrix $A$ and determine dimension of $\ker(A)$ and dimension of $\operatorname{Im}(A)$. With
$$A=\begin{pmatrix}1&-2&4&-1\\3&-1&2&-1\\-1&-3&...
0
votes
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?
0
votes
2answers
18 views
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 $&...
0
votes
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 ...
1
vote
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 ...
0
votes
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'-...
0
votes
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 $...
0
votes
0answers
34 views
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 ...
-1
votes
0answers
21 views
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
votes
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 ...
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
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
...
0
votes
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 ...
0
votes
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
vote
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 ...
0
votes
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 $F(t,y,y',..., y^{(n)})=0$ ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 = \...
0
votes
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 ...
0
votes
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
4
votes
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
votes
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:
$$\begin{array}{l}U=\{\lambda(\begin{array}{c}u_1\\\vdots\\u_n\end{...
4
votes
1answer
28 views
Question related to the trace and the commutator of matrices
Let $K$ be any field and $n\in \mathbb N$. For every $A\in M_n(K)$, define a linear form $\lambda_A: M_n(K) \rightarrow K$ by sending the matrix $M$ to $\lambda_A(M):= \operatorname{Tr}(AM)$. The map $...
2
votes
1answer
39 views
determinant of a complex matrix
Suppose we consider a complex $n\times n$ matrix $A$. It can be writen generally as $A=U+iV$, where $U$, $V$ are $n\times n$ real matrices.
Now I hope to study the determinant of the following matrix
...
4
votes
1answer
19 views
If $M$ is symmetric posdef, then all diagonal band matrices derived from $M$ are also posdef?
Let $M$ be a positive definite and symmetric matrix:
$$M = \left(\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n}\\
a_{21} & a_{22} & \cdots & a_{2 n}\\
\vdots &...
0
votes
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}...
0
votes
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 ...
0
votes
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) ...
