Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, etc. For questions specifically concerning ...
5
votes
1answer
326 views
Geometry problem from a Berkeley course
I've been trying to solve this problem proposed as part of one of the first lectures of a Berkeley linear algebra course:
"What Good is a Basis ?
The freedom to choose a basis often simplifies ...
3
votes
1answer
277 views
When are two diagonal matrices congruent?
This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
3
votes
0answers
98 views
Question about ring and module
Consider a tangent bundle with even and odd parts $T_0 + T_1$, define a space $\Omega^{k,l}_{p,q}$ consisting of (p,q)-forms taking values in $\wedge^k T_0 \otimes \wedge^l T_1$, i.e. the space of ...
1
vote
1answer
220 views
For which value(s) of parameter m is there a solution for this system
Imagine a system with one parameter $m$:
\begin{cases}
mx + y = m\\
mx + 2y = 1\\
2x + my = m + 1
\end{cases}
Now the question is: when does this system of equations have a solution?
I know how to ...
2
votes
1answer
777 views
Finding a particular solution to a non-homogeneous system of equations
If one asked to solve the set of equation below with the associated homogenous system, I'd know how to do it.
$$S \leftrightarrow \begin{cases}
3x + 5y + z = 8\\\
x + 2y - 2z = 3
\end{cases}$$
...
7
votes
1answer
183 views
Spielman's proof of graph connectivity
I use Spielman's lectures on course Spectral Graph Theory
I have few question regarding Lecture 2. The Laplacian, especially Lemma 2.3.1 (Graph connectivity). Please, help me to make it a little bit ...
3
votes
2answers
223 views
Sum and intersections of vector subspaces $U_1+U_2=(U_1 \cap U_2) \oplus W$
Let $U_1,U_2$ be vector subspaces from $\in \mathbb R^5$.
$$\begin{align*}U_1 &= [(1,0,1,-2,0),(1,-2,0,0,-2),(0,2,1,2,2)]\\
U_2&=[(0,1,1,1,0),(1,2,1,2,1),(1,0,1,-1,0)]
\end{align*}$$ ...
14
votes
4answers
2k views
Given a matrix, is there always another matrix which commutes with it?
Given a matrix $A$ over a field $F$, does there always exist a matrix $B$ such that $AB = BA$? (except the trivial case and the polynomial ring?)
1
vote
0answers
287 views
Moore–Penrose pseudoinverse reference
Given the eigendecompositions $AA^{\top}=Q \Lambda Q^{\top}$ and $A^{\top}A=P \Lambda P^{\top}$, where $\Lambda$ is a diagonal matrix (of eigenvalues) and $P$ and $Q$ are unitary eigenvectors matrices ...
2
votes
1answer
89 views
Proof: $CA = I_n$ and $AD = I_m \Rightarrow C = D$
In my school book there's an exercise:
We have three matrices $A \in \mathbb{R}^{m \times n}$ and $C, D \in \mathbb{R}^{n \times m}$.
If $CA = I$ and $AD = I$ than we can say that $C = D$.
Proof ...
1
vote
1answer
357 views
General method for factorizing matrix determinants
I'm learning how to factorize determinants of a square matrix in school, but we haven't learnt a general method to do that, besides 'creating zeros'. So I thought maybe I'll ask here if someone does ...
1
vote
1answer
146 views
matrix of linear transformation
The linear transformation $A:\mathbb{R}^2\to \mathbb{R}^2$ is given by the images of basis vectors:
$A((1,1))=(2,1)$ and $A((1,0))=(0,3)$.
Find a matrix of linear transformation $A$ in the basis ...
1
vote
1answer
264 views
Quadratic forms and prime numbers in the sieve of Atkin
I'm studying the theorems used in the paper which explains how the sieve of Atkin works, but I cannot understand a point.
For example, in the paper linked above, theorem 6.2 on page 1028 says that if ...
21
votes
2answers
2k views
What do eigenvalues have to do with pictures?
I am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article :
Can someone explain this to me ?
1
vote
2answers
129 views
Need Help Creating an Unbiased Rating System
The problem, in general terms:
At work, we're supposed to come up with an algorithm to split the expenses incurred during shipping when more than one order is being shipped on the same truck. ...
2
votes
3answers
220 views
Algebra: Orthogonal Complement
Problem
Let $V$ be a real inner product space and $U \subset V$. Show that $(U^{\perp})^{\perp}=U$.
Progress
Clearly for $x\in U$ we have that $\langle x,v \rangle=0$ for all $v \in ...
8
votes
2answers
884 views
Countable/uncountable basis of vector space
I've stumbled upon this exercise in algebra book, in chapter dealing with vector spaces' dimensions.
Prove that basis of the field of real numbers $\mathbb R$ as vector
space over the field of ...
5
votes
1answer
102 views
R is a PID , M a free R module, R-basis
We are a group of Finnish second semester undergrad students who are trying to solve old exercises from this linear algebra class for the benefit of our whole semester and even all prospective ...
1
vote
2answers
76 views
Orthogonality of vectors in $\mathbb{R}^3$
I have to show the following:
If $v, w \in \mathbb{R}^3 \setminus \{(0,0,0)\}$ such that the set of vectors orthogonal to both of them is a plane through the origin, then each is a scalar ...
4
votes
1answer
524 views
Invariants of a matrix
I'm teaching a course in physics, and I need a simple and intuitive proof that a matrix ($3\times3$, but it doesn't matter) has exactly 1 invariant which is linear in its entries, 2 that are ...
1
vote
2answers
378 views
bases for hermitian matrices
So I was working on a specific problem related to Hermitian matrices. If we let $H_n$ denote the set of n x n Hermitian matrices. We're told that $H_n$ is a real vector space under matrix addition and ...
0
votes
0answers
52 views
What are algorithms or approaches to find a convex hull on higher dimensions?
I have some background in 2D computational geometry and understand how to find a convex hull in 2D. Now I'm looking at a set of vectors with 20-some components and want to find the convex hull on ...
1
vote
1answer
64 views
Follow up question on linear transformations
So I asked a question recently:
Finding the matrix of this linear transformation
and I'm wondering something else.
$T : V \to V$ is a linear transformation. There is a vector $v \in V$ such ...
1
vote
1answer
131 views
Subspaces of linear transformation problem
I have a linear transformation $T : V \to W$, and $Z$ is a subspace of $W$. We also have $U = \{v \in V\}$ such that $ T(v) \in Z$, and we want to show that U is a subspace of V.
So basically they're ...
1
vote
2answers
118 views
Matrix multiplication
So I have a general question on matrix multiplication. I know the order of multiplication matters, except if they're invertible. So, consider something like:
$A(X+B)C = I$. If all three are ...
1
vote
1answer
128 views
Finding the matrix of this linear transformation
We're given $V$, which is an $n$ dimensional vector space. $T : V \to V$ is a linear transformation. There is a vector $v \in V$ such that $T^n(v) = 0$. We're also told that the vectors ...
2
votes
1answer
59 views
The dimensions theorem and different spaces
I'm trying to wrap my head around the meaning of different dimensions in lin alg. Consider the matrix:
$$\left[ \begin{array}{ccc}
1 & -1 \\
2 & -2\\
3 & -3
\end{array} \right] $$
...
1
vote
2answers
152 views
Projections onto ranges/subspaces
I'm stuck on a review problem.
Consider the matrix:
$$\left[ \begin{array}{ccc}
-1 & 1 \\
1 & 1\\
2 & 1
\end{array} \right] $$
I'm asked to find a matrix $P$ which projects onto ...
2
votes
3answers
56 views
How to solve this particular linear system?
So I have this linear system:
$$
\begin{align*}
-u + v &= y_1\\
u + v &= y_2 \\
2u + v &= y_3
\end{align*} $$
After doing gaussian elimination, I get:
$$
\begin{align*}
u = y_3 - ...
0
votes
3answers
127 views
A basis of a vector space
Let $v_1, \ldots, v_n$ be a set of vectors in a vector space $V$. Show that $v_1, \ldots, v_n$ is a basis of $V$ if and only if for any non-zero linear function $f$ on $V$ there is a vector $v$ in ...
1
vote
1answer
217 views
Power of a matrix using Sylvester's Formula
I have been thinking about this question and I'm really confused, I have gone through past solutions and I really understand those, but this, I don't understand. I'm to use Sylvester's formula to find ...
2
votes
3answers
167 views
Finite dimensional subspaces of a linear space
Suppose $V$ is an infinite dimensional vector space. I do not want to assume the axiom of choice, so I will define a vector space $V$ to be infinite dimensional if there is a proper subspace ...
3
votes
2answers
120 views
Question on finding eigenvalues of another linear transformation
I have a quick question on finding eigenvectors for a linear transformation. I'm given:
$T(A) = A^t$ where $A = M_2$ i.e. a $2 \times 2$ matrix consisting of real numbers.
So the general approach is ...
0
votes
3answers
177 views
Verify ten axioms of a tangent space
So I know that tangent space as a set of linear approximation of all tangent vectors. And consequently, a tangent vector can be defined at a point in a vector space, as a order of $n$-tuples ...
1
vote
1answer
317 views
Linear Algebra: Dual Basis Problem
Problem
Let $V$ be the vector space of all polynomial functions $p$ from $\mathbb{R}$ to $\mathbb{R}$ which have degree two or less.
Define three linear functionals on $V$ by ...
4
votes
1answer
866 views
How to find eigenvalues of this linear transformation?
I have a practice problem on linear transformations that I'd like help on. I have to find the eigenvalues and eigenvectors of
$$T(ax^2 + bx + c) = bx^2 + cx .$$
So I know the general idea is that ...
1
vote
1answer
48 views
Diagonalizing matrices and powers
I'm wondering about the approach to another problem.
$B = \left( \begin{array}{ccc}
a & 2a & 3a \\
4b & 5b & 6b \\
7c & 8c & 9c \end{array} \right) $
and I want to find ...
1
vote
2answers
73 views
Questions on rank and invertability
I'm working on some practice problems, and I'd like some feedback.
We have a n x n square matrix $A$ with rank $r$. There are other matrices $B$ and $C$ and $AB=AC$. I'm asked to find the maximal ...
0
votes
1answer
120 views
Conceptual question on dimension theorem
$\newcommand{rank}{\operatorname{rank}}\newcommand{nullity}{\operatorname{nullity}}$
Sorry if this question is too basic, but I'm wondering something. So the Fundamental Theorem of Linear Algebra says ...
4
votes
2answers
280 views
Etymology of the word “isotropic”
Given a quadratic form $q : V \rightarrow k$, a nonzero vector $v \in V$ is said to be isotropic if $q(v) = 0$. Any subspace of $V$ containing such a vector is also said to be isotropic, and the ...
0
votes
1answer
48 views
PageRank using Power Extrapolation
I am trying to understand "Computing PageRank using Power Extrapolation" by Taher Haveliwala, Sepandar Kamvar, Dan Klein, Chris Manning, and Gene Golub from Stanford University.
The paper I'm ...
1
vote
3answers
131 views
Conventions for complex inner product: does it affect formulae?
I'm writing a linear algebra exam next week and it's come to my attention that the prof that designed the test uses a different convention for complex inner product than the one my prof taught me.
I ...
1
vote
1answer
87 views
$V$ is $\mathbb K$ vector space; Basis $B$; $A \subset B$; Show $(B \backslash A) \cup M$ is a basis
sorry it's me again!
Let $V$ be a $\mathbb K$ vector space with finite basis $B$ and $M \subset V$ is a finite linear independent subset.
Show that a subset $A \subset B$ exists, so that $(B ...
-1
votes
1answer
95 views
Vector Spaces Question
A) Is it true that always the group $\{(Z_1,Z_2) \in \mathbb{C}^2\;|\;Z_2 \in \mathbb{R}\}$ is a subspace of $\mathbb{C}^2$ as a vector space over $\mathbb{C}$?
B) Is it true that always the group ...
2
votes
2answers
259 views
Can all matrices be orthogonalized?
I helped a buddy of mine do some MATLAB homework where you had to orthogonalize a matrix via Gram-Schmidt. I wrote a test function called isorthog that returned ...
1
vote
0answers
59 views
Stable and efficient projection onto subspace along another subspace
Suppose we are given the euclidean space $\mathbb R^{n+m}$ with the decompositin $\mathbb R^n = V \oplus W$, which we however do not expect to be orthogonal.
Let us describe the matrix $P$ that ...
2
votes
2answers
43 views
Operator Commutativity $\hat{L}_{z}$; $\hat{p}^{2}$; $[\hat{p}^{2}, \hat{L}]$
Define:
$$\hat{p}^{2}= \hat{p}^{2}_{x}+\hat{p}^{2}_{y}+\hat{p}^{2}_{z}= -\hbar^{2} \Delta$$
$$\hat{L}_{z}= xp_{y}-yp_{x}= -i\hbar \big(x\frac{\partial}{\partial y} - y \frac{\partial}{\partial x} ...
0
votes
1answer
58 views
Solution of a system of second order algebraic equations in complex numbers
What is the simplest solution for this set of equations:
$
\sum_{i=1,3,5,..}^{N-1} \left | x_i \right |^2=c_1,\
\sum_{i=2,4,6,..}^{N} \left | x_i \right |^2=c_2,\
\sum_{i=1,3,5,..}^{N-1} ...
1
vote
1answer
147 views
Linear Span properties
Let $V$ be a $\mathbb K$ vector space, $M_1$ and $M_2$ subsets of $V$. ($[M] =$ linear span of $M$) Show that:
a) $[M_1 \cup M_2] = [[M_1 \cup M_2]]$ and specially $[[M]] = [M]$.
What I've ...
1
vote
2answers
70 views
Why does counting companion matrices count conjugacy classes?
A method of counting conjugacy classes of say $GL(n,\mathbb{F}_q)$ is to count the possible companion matrix blocks in the rational canonical form by counting the possible monic irreducible ...
