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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.
1
vote
0answers
21 views
Ring homomorphisms from $\mathbb{R}$ to another unital ring $S$.
We know that there is only one non-trivial ring homomorphism from $\mathbb{Z}$ or $\mathbb{Q}$ to another unital ring $S$.
What’s more,when we consider the automorphism of $\mathbb{R}$,it is unique ...
1
vote
1answer
11 views
Is there an easy way to remove scale from a squared linear transformation matrix
Given a linear transformation matrix $A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}$, I know that one can use SVD or QR decomposition ...
0
votes
0answers
17 views
LU decomposition of matrix product
Let $A_1=L_1U_1$ and $A_2=L_2U_2$ be two matrices with their respective LU-factorizations ($L_i$ is lower triangular and $U_i$ upper triangular).
Is it possible to obtain the LU decomposition of the ...
0
votes
0answers
8 views
Determining the matrix in an integration by substitution context
I am working on a problem in which integration by substitution is a step. In particular, we are substituting $(u,v) \mapsto (t,z)$ s.t.
$$
u = tz \\
v = (1-t)z
$$
I know that conceptually when we ...
1
vote
2answers
35 views
State true or false ( I am not sure what i did wrong)
For 𝐮,𝐯 ∈ ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮+𝐯‖.
The dot product of two vectors is a vector.
For 𝐮,𝐯∈ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮‖+‖𝐯‖.
A homogeneous system of linear equations with more equations than ...
0
votes
0answers
18 views
If two operators with all eigenvalues real numbers commute then both are triangularizable!
Let $A,B:E\to E$, where $E$ is a finite dimensional vector space, be two linear operators such that all of the eigenvalues are real numbers.
If $AB=BA$, prove that there exists a basis in which ...
2
votes
0answers
11 views
Non existence of skew-symmetric matrices with the difference of their square is diagonal of some type
By using Maple I can show that there is no skew-symmetric matrices $J_1$ and $J_2$ of order 4 satisfying
$$J_1^2-J_2^2=\left(\begin{array}{cccc}-\frac12Trace(J_1^2)&0&0&0\\
0&\...
0
votes
0answers
17 views
Distance Between 2 Points in Matrix Form
Let $$ \begin{bmatrix}-m_1 &1\\-m_2 &1\\ \vdots & \vdots\\-m_n &1\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix} c_1\\ c_2\\ \vdots\\ c_n\end{bmatrix} $$
and
$$ \...
1
vote
0answers
11 views
Projection on to subspace with positive constraint?
I have an underdetermined system of 2 equations in n variables and a "starting point" $s$. I need to find the point satisfying the equations which is closest to the starting point. However, I have the ...
-3
votes
2answers
24 views
${\bf a}= 2 {\bf i} + 2 {\bf j} + 27 {\bf k}$ and ${\bf b} = {\bf i}+ \lambda {\bf j} + \mu {\bf k}$ and ${\bf a} \times {\bf b} = {\bf 0}$ [on hold]
Given the above conditions, find $\lambda$ and $\mu$.
0
votes
1answer
20 views
How to solve 3 linear equations in three variables using cross multiplication method?
How to solve 3 linear equations in 3 variables using cross multiplication method? I have no problem in solving these equations using substituting. However, how do I solve these using cross ...
0
votes
0answers
13 views
Rotation of vector by rotation matrix
Assume the following expression
$$
\begin{bmatrix}
a_1^* \\
a_2^*
\end{bmatrix}
=
\begin{bmatrix}
\cos(45) & - \sin(45) \\
\sin(45) & \cos(45)
\end{bmatrix}
\begin{bmatrix}
a_1 \\
a_2
\end{...
0
votes
2answers
30 views
Does a right inverse of a linear map, have to be a linear map? [duplicate]
Let T be a linear map from $\mathbb{R}^3 \rightarrow \mathbb{R}^2$. Let K be a right-inverse of T. Does K have to be a linear map/transformation.
3
votes
4answers
182 views
Must every right-inverse of a linear transformation be a linear transformation?
Let T be a linear transformation $\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Let S be the right-inverse of T. Does S have to be linear transformation?
Thanks in Advance.
0
votes
1answer
14 views
Can a linear transformation be expressed by constants?
Let T be a linear transformation from $\mathbb{R}^4 \rightarrow \mathbb{R}^5$.
$$T
\begin{pmatrix}
x_1\\
x_2\\
x_3\\
x_4
\end{pmatrix}
=
\begin{pmatrix}
x_1\\
x_2\\
x_3\\
x_4\\
1
\end{pmatrix}
$$
Is ...
0
votes
1answer
24 views
Projecting into a plane
Given vectors are $v_1 = (1,-2,1,3) $ , $v_2=(0,3,1,-1) $ and $v_3=(1,1,-1,-1) $ in $ R^4$ with the standard inner product. The question is how to find the orthogonal projection of $v_3$ on the ...
0
votes
0answers
12 views
Basis complement of image of sparse matrix
I have a map of vector space $R:T\to M$, and I'm interested in the quotient $E=M/R(T)$. In particular I want to construct a section $s:E\to M$. Theoretically this is not very difficult to solve, but I ...
-1
votes
1answer
26 views
Given the base $B = \{(1,1,a) (0,b,-1) (c,3,0)\}$, find $a,b,c$ [on hold]
Given the base $B = \{(1,1,a) (0,b,-1) (c,3,0)\}$, find $a,b,c$
So that is that, I know that if they are a basis, then all must be independent, but at the moment I try to solve it, it gives me ...
0
votes
0answers
10 views
Relative position of hyper planes in $\mathbb{K}^5$ [on hold]
Considering the affine space $\mathbb{K}^5$, given the subspaces:
$$X:x-1=y=z-1=t=0$$
$$X’:x=y-1=w-5=0$$
Study their relative position and calculate the set of Cartesian (implicit) equations for $X+X’...
2
votes
1answer
18 views
Equivalence relation on matrices
Consider $M_{n\times n}$ the $n\times n$ matrices over some field $F$. Define an equivalence relation $A\sim B$ if there is invertible $C$ such that $A=CB$. What are the equivalence classes of $\sim$ ...
1
vote
2answers
55 views
Relation between $\operatorname{trace}(AB)$ and $\operatorname{trace}(ABAB)$
Let $A$ and $B$ two square positive definite matrices of order $k$. Is there any relation between $\operatorname{trace}(AB)$ and $\operatorname{trace}(ABAB)$ for generic $k$? So far I have found that $...
0
votes
1answer
16 views
The various definitions of adjoint
Mathematics appears to have many different definitions of adjoint. There are adjoint representations on Lie algebras, adjoints of a matrix, adjoint actions on the space of linear transformations. Are ...
0
votes
1answer
28 views
Show that $\{x^4+x^3-1,x^3-x,x^2+1\}⊂E$ is a generating set of span($E$).
I'm given the following set of polynomials:
$$E:=\{x^4+x^2+x,x^4+x^3-1,x^3-x,x^2+1\}$$
I know that $E$ is linearly dependent because when $\alpha_1 = -1$, $\alpha_2=1$, $\alpha_3=-1$ and $\alpha_4=1$...
3
votes
2answers
49 views
How to determine the base of $\ker\phi$ for polynomial function?
Given is a base defined as $$B:=(x\mapsto1,x\mapsto x,x\mapsto x^2,x\mapsto x^3 ,x\mapsto x^4)$$ A set V defined as
$$V:= \{ f: \mathbb{R} \mapsto \mathbb{R}\ |\ \exists\ {a_0},...{a_4} \in \mathbb{R}\...
1
vote
2answers
32 views
If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$?
I'm a student and I've just read the Characteristic polynomial on Wiki.
I have a feeling that:
If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$
thanks to the Matrix calculator I've ...
0
votes
1answer
23 views
Finding minimum of penalty-approximated quadratic problem
Find the minimum of the following quadratic function
$$ f_{\alpha}(x) := \frac{1}{2} x^T H x + c^T x + \frac{\alpha}{2}(b^Tx)^2 $$
where matrix $H$ is symmetric and positive definite, and $\...
0
votes
0answers
7 views
Computing the shifted symmetric polynomial $s^*_{(1^2)}(x_1,x_2)$
By definition,
Let $\lambda=(\lambda_1,...,\lambda_n)$ be a partition with $l(\lambda)\leq n$. We define the shifted Schur polynomial in $n$ variables corresponding to $\lambda$ as
\begin{equation*}
...
4
votes
2answers
70 views
Can a matrix $A$ commute with $e^B$ without commuting with $B$?
As in the title. Is it possible that $[A,B]\neq0$, but $[A,e^B]=0$? I tried expanding the exponential and using $[A,B^n]=\sum_k {n\choose k} B^{n-k}[A,B]B^k $ but this doesn't seem to give any insight....
0
votes
3answers
26 views
Prove for complex matrices $A, B$ and $C$, if $AB = BC$ than for each natural number $k$, $\\\\A^kB = BC^k $
in this question and its prove, it is stated that:
For complex matrices $A, B$ and $C$, if $AB = BC$ than for each natural number $k$, $\\\\A^kB = BC^k$
I can't see why. Any help would be ...
1
vote
1answer
34 views
determinant of changing of basis matrix
Let $B=\{f_{1},…,f_{n},g_{1},…g_{n}\}$ the order orthonormal basis of $2n$-dimensional and $B′=\{f_{1}+g_{1},f_{1}−g_{1},…,f_{n}+g_{n},f_{n}−g_{n}\}$ the other order basis. How can I compute ...
0
votes
0answers
7 views
Relation between leading eigenvlaue and eigenvector of individual blocks with the leading eigenvector of non-negative symmetric block matrices
For a 2 by 2 block non-negative symmetric matrix $\mathcal{M}$,
$\mathcal{M}=
\left[
\begin{array}{c|c}
\mathcal{A} & \mathcal{C} \\
\hline
\mathcal{C}^T & \mathcal{B}
\end{array}
\right]$
...
0
votes
0answers
24 views
Tangent space of matrix group
In Andrew Baker's book 'Matrix Groups', he as defined tangent space to $G$, a matrix group, at $U \in G$ as $$T_U G =\{\gamma ' (0) \in M_n(\mathbb K) : \gamma \ \text{is differentiable curve in} \ G ...
0
votes
1answer
13 views
Proving a specific $min$ function is equivalent to solving $Ax-b$
The homework question asks to prove that
$min_{x\in\mathbb{R}} {f(x) = 1/2<Ax,x>-<b,x>}$
is equivalent to solving a linear system $Ax-b$.
The hint the professor gave is to recite the ...
1
vote
0answers
25 views
What is the meaning of Motzkin's theorem?
Theorem: Let $A$ and $C$ be two matrices. The system of linear inequalities $Ax<0$ and $Cx \leq 0$ has a solution iff the following equation in $\lambda$ and $\mu$ does not have a solution$$A^T \...
1
vote
1answer
24 views
Set of independant vectors spanning a plane
Suppose I have a set of $n$ independent vectors in $\mathbb{R}^{m}$ where $2 < n < m$
My understanding is that these vectors do not form a basis as $n < m$, but do they span a plane in $\...
0
votes
0answers
13 views
Checking if lines intersect or skew from directions ratios and a point on each line
Well , this is what's in my book
$DR_1(m1,m2,m3)$ with $A(a1,a2,a3)$
$DR_2(n1,n2,n3)$ with $B(b1,b2,b3)$
\begin{vmatrix} m1 & m2 & m3 \\ n1 & n2 & n3 \\ a_1-b_1& a_2-b_2 & ...
1
vote
1answer
25 views
What it means for linear functionals to be linearly independent?
Suppose we have the cannonical basis in dimension $3$, I guess three linear functionals are:
$$f_1(x,y,z)=x\\f_2(x,y,z)=y\\f_3(x,y,z)=z$$
Are they linearly independent? I'm confused because if I try ...
0
votes
1answer
18 views
Find a linear map $T : \mathbb{M}_{1 \times 4}(\mathbb{R}) \rightarrow \mathbb{R}^5$ with given Kernel
Find a linear map $T : \mathbb{M}_{1 \times 4}(\mathbb{R}) \rightarrow \mathbb{R}^5$ with Kernel generated by
$$ \vec{v_1} = ~[1 ~~2~~3~~4];~~ \vec{v_2} =~[0~~1~~1~~1] $$
Thoughts on this ...
-2
votes
1answer
46 views
An $n \times n$ matrix $A$ is called skew-symmetric if $A^T = −A$. What values of a, b, c, and d now make the following matrix skew-symmetric? [on hold]
Let $$ A=\left(
\begin{matrix}
d & 8a-c & 8a+2b \\
a & 0 & 8-5d \\
a+5b & c & 0 \\
\end{matrix}
\right)
$$
Let $$ A^T=\left(
\begin{matrix}
d & ...
1
vote
3answers
45 views
$T : \mathbb{R}^3 \rightarrow \mathbb{R}^2$ and $S : \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Show that $S \circ T$ is not invertible.
Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^2$ and $S : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be linear transformations. Show that $S \circ T$ is not invertible.
Thoughts on this problem:
$S$ can ...
4
votes
0answers
41 views
Kronecker Product Interpretation
The algebraic expression for a Kronecker product is simple enough. Is there some way to understand what this product is?
The expression for matrix-vector multiplication is easy enough to understand. ...
0
votes
2answers
40 views
Let A, B be two matrices, which satisfy Det(AB) $\neq$ Det(BA), prove rank(A) = rank(B) = 4
Let $A$ $\in$ $\mathbb{R^{4 x5}}$ and $B$ $\in$ $\mathbb{R^{5 x4}}$ which satisfy det(A$\cdot$B) $\neq$ det(B$\cdot$A). Prove rank(A) = rank(B) = $4$.
So far I've tried separating the cases where one ...
-1
votes
1answer
25 views
how to choose eigenvectors for repeated eigenvalue when we do spectrum decomposition?
I know that different eigenvectors from different eigenspace are automatically orthogonal.
My question is:
Suppose we are doing spectrum decomposition to a 3x3 symmetric matrix and
we have only ...
1
vote
2answers
46 views
Zero of a polynomial and divisbility
I have a polynomial $p(x)$, $deg(p) = n$. I know that $\alpha$ is a zero of $p(x)$. Then $(x-\alpha)|p(x)$. Is it wrong to say that $(x-\alpha)^m|p(x)$, $m \in \mathbb{N}, m>1 $?
0
votes
0answers
18 views
Argument that the picture of P is in U
I should argument that this is $P^2 = P$ because the picture of P is in U.
can I argument that if I define this vector $v ∈ R^n and \ n ∈ N$
so $v = \left(\begin{matrix}
a
\\b
\\c
\end{matrix}\...
1
vote
0answers
11 views
Derive Hessian inverse update using Sherman-Morrison in Quasi Newton Method
From Nocedal's Numerical Optimization book, the Hessian approximation equation is given using Symmetric Rank 1 (SR1) formula as follows:
$$B_{k+1} = B_k + \frac{(y_k - B_ks_k)(y_k - B_ks_k)}{(y_k - ...
0
votes
1answer
19 views
Systematic approach to find base of vectorspace given its elements' traits
I'm trying to find a base for a vector space that's given as a set with certain traits. Take this example:
Let $V$ be an $\mathbb{R}$-vector space with
$$ V := \left\{ (a, b, c, d) \in \mathbb{R}^4 : ...
1
vote
3answers
30 views
How to find basis of linear subspace $V$ when $V$ contains all polynomials with the degree up to 4?
I have been given the following definition
$V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there are } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $x \in \mathbb{R}\}$...
0
votes
1answer
22 views
Questions about singular value decomposition
I'm still having trouble understanding the separation between diagonalization and singular value decomposition. I am working with complex matrices so for an arbitrary matrix, I am able to write it as $...
0
votes
2answers
40 views
If $A^n=0$, then $\left(BAB^{-1}\right)^n=0$
If $A^n=0$, then $\left(BAB^{-1}\right)^n=0$. (Assume everything that's necessary for the products to make sense.)
Proof:
$$\left(BAB^{-1}\right)^n=\underbrace{\left(BAB^{-1}\right)\left(BAB^{-1}\...
