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 ...
49
votes
16answers
8k views
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$ where $I$ is identity matrix. Show that $BA = I$. I do not understand anything more than the following.
Elementary row operations.
Linear ...
-1
votes
3answers
155 views
Solution of 3 equations in 3 unknowns
Find the value of $c$ which makes it possible to solve:
$$u+v+2w=2,$$
$$2u+3v-w=5,$$
$$3u+4v+w=c$$
131
votes
9answers
8k views
What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
4
votes
3answers
567 views
A vector space over $R$ is not a countable union of proper subspaces
I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field is not a union of fewer than $|\mathbf{F}|$ proper subspaces" and ...
13
votes
11answers
4k views
Where to start learning Linear Algebra?
I'm starting a very long quest to learn about math so I can program games. I'm mostly a coorporate developer, and it's somewhat boring and non exiting. When I began my career, I chose it because I ...
38
votes
2answers
2k views
Axiom of choice and automorphisms of vector spaces
A classical exercise in group theory is "Show that if a group has a trivial automorphism group, then it is of order 1 or 2." I think that the straightforward solution uses that a exponent two group is ...
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?)
7
votes
3answers
6k views
Union of two vector subspaces not a subspace?
I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
5
votes
3answers
910 views
Sylvester's determinant identity
Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ det(I+AB)=det(I+BA), $$ where in the first case $I$ denotes the $m\times m$ ...
59
votes
2answers
14k views
What is the intuitive relationship between SVD and PCA
Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional dataset into fewer dimensions while retaining important ...
33
votes
4answers
3k views
Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
24
votes
3answers
2k views
Why are vector spaces not isomorphic to their duals?
Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$).
I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the ...
14
votes
9answers
4k views
How to show $\det(AB) =\det(A)\det(B)$
Given two square matrices $A$ and $B$. How do you show $\det(AB) = \det(A)\det(B)$ where $\det(\cdot)$ is determinant of the matrix
12
votes
6answers
600 views
How to solve this recurrence relation? $f_n = 3f_{n-1} + 12(-1)^n$
How to solve this particular recurrence relation ?
$$f_n = 3f_{n-1} + 12(-1)^n,\quad f_1 = 0$$
such that $f_2 = 12, f_3 = 24$ and so on.
I tried out a lot but due to $(-1)^n$ I am not able to ...
1
vote
2answers
3k views
Simultaneous diagonalization
Let $V$ be a vector space from a finite dimension and let $T,S$ linear diagonalizable transformations from $V$ to itself. I need to prove that:
a. If $TS=ST$ so every eigenspace $V_\lambda$ of $S$ ...
28
votes
3answers
959 views
linear algebra over a division ring vs. over a field
When I was studying linear algebra in the first year, from what I remember, vector spaces were always defined over a field, which was in every single concrete example equal to either $\mathbb{R}$ or ...
11
votes
2answers
3k views
matrix multiplication: interpreting and understanding the multiplication process
I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link:
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/
It ...
8
votes
1answer
739 views
Probability that a random binary matrix is invertible?
What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible?
Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$.
Is there an explicit formula as a ...
3
votes
3answers
340 views
What is step by step logic of pinv (pseudoinverse)?
So we have a matrix $A$ size of $M \times N$ with elements $a_{i,j}$. What is a step by step algorithm that returns the Moore-Penrose inverse $A^+$ for a given $A$ (on level of ...
1
vote
2answers
164 views
Determining eigenvalues, eigenvectors of $A\in \mathbb{R}^{n\times n}(n\geq 2)$.
Let $a$ and $b$ be distinct nonzero real numbers and let $A\in \mathbb{R}^{n\times n}(n\geq 2)$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$.
Determine all ...
12
votes
4answers
384 views
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
I was reviewing some matrices and found this interesting
if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
11
votes
3answers
410 views
For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?
True\False?
For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$.
I know that every complexed matrix has a Jordan form matrix $J$ such that $P^{-1}CP=J$, ...
3
votes
2answers
357 views
Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?
I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these.
It seems to be a well known result that when you take the eigenvectors of A as a basis and ...
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 ?
10
votes
2answers
5k views
matrices commute=common set of eigenvectors?
I've come across a paper that mentions this as a fact...where can I find the proof?
14
votes
5answers
945 views
$\sin(A)$, where $A$ is a matrix
If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
16
votes
2answers
4k views
Why is the eigenvector of a covariance matrix equal to a principal component?
If I have a covariance matrix for a data set and I multiply it times one of it's eigenvectors. Let's say the eigenvector with the highest eigenvalue. The result is the eigenvector or a scaled ...
3
votes
7answers
676 views
How to calculate the following determinants
How do I calculate the determinant of the following $n\times n$ matrices
$
\left[
\begin {matrix}
0 & 1 & \ldots & 1 \\
1 & 0 & \ldots & 1 \\
\vdots & \vdots & ...
19
votes
6answers
1k views
Beautiful identity: $\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$
Let $m,n\ge 1$ be two integers. Prove that
$$\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$$
where $\delta_{mn}$ stands for the Kronecker's delta.
Note: I put the tag "linear ...
4
votes
4answers
2k views
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
Let A and B be two matrices which can be multiplied.
Then $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$
I proved $\operatorname{rank}(AB) \leq ...
12
votes
4answers
791 views
Similar Matrices
Given a field $F$ and a subfield $K$ of $F$ . Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$, is it true that
if $A$ is similar to $B$ in $F^{n\times n}$ then ...
8
votes
4answers
1k views
Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]
Possible Duplicate:
Units and Nilpotents
Hi all wondering if I could get a bit of help with this, given $A^{2012}=0$ prove $(A+I)$ is invertible and find an expression for $(A+I)^{-1}$ in ...
3
votes
2answers
295 views
If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$
If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
7
votes
7answers
484 views
Determinant of a specially structured matrix
I have the following $n\times n$ matrix:
$$A=\begin{bmatrix}a&b&\cdots&b\\b&a&\cdots&b\\\vdots& &\ddots&\vdots\\b&\cdots&b&a\end{bmatrix}$$
where $0 ...
14
votes
5answers
4k views
Importance of rank of a matrix
What is the importance of rank of a matrix ?
I know that rank of a matrix is the number of linearly independent rows/columns (whichever is smaller). Why is it a problem if a matrix is rank ...
13
votes
2answers
1k views
Why is the ring of matrices over a field simple?
Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal?
Thanks for any clarification, and this is ...
10
votes
6answers
2k views
Why determinant of a 2 by 2 matrix is the area of a parallelogram?
Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$
be a two by two matrix where the first row of $A$ is $a, b$ and the second row of $A$ is $c, d$. How could we show that $ad-bc$ is the area of ...
9
votes
3answers
831 views
An orthonormal set cannot be a basis in an infinite dimension vector space?
I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every ...
6
votes
6answers
1k views
Prerequisites/Books for Linear Algebra
What mathematical knowledge do I need to begin studying linear algebra? In particular, how much calculus do I need to know?
Also, do you have a favorite linear algebra book you can recommend?
4
votes
3answers
929 views
When are minimal and characteristic polynomials same?
Assume that we are working over a complex subspace W of dimension n. When would an operator on this subspace have the same characteristic and minimal polynomial? I think the easy case is when the ...
12
votes
2answers
774 views
necessary and sufficient condition for trivial kernel of a matrix over a commutative ring
In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
7
votes
2answers
717 views
If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$
If $A\in M_{n}$ has $n$ distinct eigenvalues and if $A$ commutes with a given matrix $B\in M_{n}$, how can I show that $B$ is a polynomial in $A$ of degree at most $n-1$? I think first I need to show ...
2
votes
4answers
280 views
A be a $3\times 3$ matrix over $\mathbb {R}$ such that $AB =BA$ for all matrices $B$. what can we say about such matrix $A$
Let $A$ be a $3\times 3$ matrix over $\mathbb {R}$ such that $AB =BA$ for all matrices $B$ over $\mathbb {R}$ then what can we say about such matrix $A$.
or such matrix $A$ must be orthogonal matrix? ...
0
votes
1answer
775 views
Finding Bases for Kernel and Image of Linear Transformation
I'm trying to solve a linear transformation problem.
Let $ \psi: \mathbb R_3 [x] \to \mathbb R_4 [x] $ be defined by $ \psi : p(x) \mapsto x^4 p(1/x)+p(x)$
Q) Show that $\psi$ is a linear ...
7
votes
1answer
193 views
Is there a non-trivial example of a $\mathbb Q-$endomorphism of $\mathbb R$?
$\mathbb R$ is an uncountably dimensional vector space over $\mathbb Q.$ We can define as many endomorphisms of this vector space as we want by picking their values on the elements of the basis. ...
6
votes
2answers
468 views
Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$
I had my exam of linear algebra today and one of the questions was this one.
Given $ A \in \mathbb{R}^{n \times n}$, prove that:
$$\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A.$$
...
5
votes
4answers
273 views
How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying ...
2
votes
3answers
562 views
I don't understand why the inverse is this?
my question is related to matrix inverting and Hill cipher(you don't have to know what it is to help me)
My teacher gave me an example. First we have a matrix (the key matrix) that multiplied by a ...
1
vote
2answers
185 views
Groups/Linear maps
Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have?
$\text{GL}_n(\mathbb{F}_3)$
$\text{SL}_n(\mathbb{F}_3)$
Here GL is the general ...
6
votes
2answers
167 views
Representation of a linear functional in vector space
In the book Functional Analysis,Sobolev Spaces and Partial Differential Equations
of Haim Brezis we have the following lemma:
Lemma. Let $X$ be a vector space and let
$\varphi, \varphi_1, \varphi_2, ...
