Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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58
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17answers
12k 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 ...
176
votes
11answers
15k 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 ...
28
votes
14answers
16k views

Where to start learning Linear Algebra?

I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because ...
-1
votes
3answers
358 views

Solution of 3 equations in 3 unknowns [closed]

Find the value of $c$ which makes it possible to solve: $$u+v+2w=2,$$ $$2u+3v-w=5,$$ $$3u+4v+w=c$$
18
votes
4answers
13k 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?
6
votes
3answers
983 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 ...
17
votes
5answers
3k 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?)
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 ...
13
votes
6answers
929 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 ...
85
votes
2answers
30k 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 ...
39
votes
4answers
5k 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?
17
votes
5answers
3k views

What is a good book to study linear algebra?

I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a ...
12
votes
8answers
3k 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 ...
7
votes
3answers
2k views

When are minimal and characteristic polynomials the same?

Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial? I think the easy case is when the ...
6
votes
3answers
2k 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 $m\times m$ ...
43
votes
6answers
18k views

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
33
votes
3answers
3k 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 ...
17
votes
2answers
6k views

Motivation behind Definition of Matrix Multiplication

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 ...
18
votes
10answers
10k 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
15
votes
2answers
8k views

matrices commute=common basis of eigenvectors?

I've come across a paper that mentions this as a fact...where can I find the proof?
11
votes
1answer
1k 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 ...
4
votes
7answers
826 views

How to calculate the following determinants (all ones, minus $I$)

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 & ...
2
votes
2answers
5k 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 $S$ ...
13
votes
3answers
501 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 $P^{-1}CP=J$, ...
2
votes
2answers
229 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 ...
35
votes
4answers
1k 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 ...
26
votes
2answers
10k 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 ...
12
votes
2answers
2k views

Matrix is conjugate to its own transpose

Mariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose. I spent quite a bit of time on it now, and still could not prove it. At the ...
19
votes
2answers
3k 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 ...
6
votes
2answers
1k views

How to show if $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$?

Statement: If $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$ and vice versa. One way of the proof. We have $B(B^{-1}A ) B^{-1} = AB^{-1}. $ Assuming $ ...
16
votes
5answers
1k views

Similar matrices and field extensions

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 $F^{n\times n}$ ...
13
votes
4answers
518 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 ...
6
votes
2answers
389 views

The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
3
votes
2answers
365 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 $U_1$, $U_2,\ldots,U_n$?
4
votes
4answers
2k views

The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial in 1 variable over a field $F$ plays an important role in understanding the structure of finite dimensional $F[x]$-modules. It is an important fact that the ...
19
votes
7answers
4k 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 ...
20
votes
5answers
10k views

How do I tell if matrices are similar?

I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ ...
16
votes
2answers
1k 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 ...
10
votes
7answers
829 views

A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
23
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 ...
13
votes
2answers
2k views

History of dot product and cosine

The fact that the dot product and the cosine of the angle between two vectors are mutually computable is easy to show (see the two sides in the two answers at Dot product in coordinates). But looking ...
7
votes
2answers
813 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: $$\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A.$$ ...
6
votes
2answers
997 views

Does the set of matrix commutators form a subspace?

The following is an interesting problem from Linear Algebra 2nd Ed - Hoffman & Kunze (3.5 Q17). Let $W$ be the subspace spanned by the commutators of $M_{n\times n}\left(F\right)$: $$C=\left[A, ...
4
votes
3answers
743 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 ...
9
votes
4answers
3k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
8
votes
4answers
2k views

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
2
votes
3answers
874 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 ...
7
votes
7answers
554 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 ...
25
votes
2answers
3k 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 ?
20
votes
5answers
28k views

Inverse of the sum of matrices

I have two square matrices - $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case ...