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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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 ...
87
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 ...
69
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18answers
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Is there another simpler method to solve this elementary school math problem?

I am teaching an elementary student. He has a homework as follows. There are 16 students who use either bicycles or tricycles. The total number of wheels is 38. Find the number of students using ...
59
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17answers
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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 ...
53
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3answers
2k views

Cute Determinant Question

I stumbled across the following problem and found it cute. Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant: ...
53
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7answers
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What exactly are eigen-things?

Wikipedia defines an eigenvector like this: An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original vector at most ...
43
votes
6answers
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What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
42
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12answers
13k views

Why study linear algebra?

Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study ...
41
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4answers
4k views

Why do we care about dual spaces?

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
39
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4answers
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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?
38
votes
2answers
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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 ...
37
votes
19answers
4k views

What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
37
votes
4answers
1k views

How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
36
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4answers
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If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?

It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true? If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? Edit: Addionally, what ...
36
votes
2answers
1k views

Polynomial equations $p(A, B) = 0$ for matrices that ensure $AB = BA$

Let $k$ be a field with characteristic different from $2$, and $A$ and $B$ be $2 \times 2$ matrices with entries in $k$. Then we can prove, with a bit art, that $A^2 - 2AB + B^2 = O$ implies $AB = ...
35
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4answers
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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 ...
34
votes
2answers
5k views

What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the ...
34
votes
3answers
2k views

Eigenvalue problem: Prove that all of the eigenvalues of $A$ are $1$.

Here's a cute problem that was frequently given by the late Herbert Wilf during his talks. Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. ...
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 ...
33
votes
1answer
347 views

Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which ...
30
votes
1answer
2k views

Is the following matrix invertible?

$$ \begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$ Clearly its ...
30
votes
2answers
687 views

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
29
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6answers
2k views

The milk sharing problem

I found a book with math quizzes. It was my father's when he was young. I encountered a problem with the following quiz. I solved it, but I wonder, is there a faster way to do it? If so, how can I ...
29
votes
1answer
360 views

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
28
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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 ...
28
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2answers
796 views

Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$?

Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any ...
27
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2answers
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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 ...
26
votes
8answers
1k views

Can the matrices $A$ and $I+A$ have the same determinant?

Let $A\in\mathbb R^{n\times n}$ be an arbitrary matrix. Can $A$ and $I+A$ have the same determinant, if not how to prove it? Furthermore, can $A$ and $I+A$ have the same eigenvalues?
26
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4answers
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Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields

Some days ago, I was thinking on a problem, which states that $AB-BA=I$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ denotes ...
26
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6answers
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Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...
26
votes
3answers
533 views

Square matrices satisfying certain relations must have dimension divisible by $3$

I saw this tucked away in a MathOverflow comment and am asking this question to preserve (and advertise?) it. It's a nice problem! Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with ...
25
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3answers
1k views

How to find the determinant of this matrix?

Today at my linear algebra exam, there was this question that I couldn't solve. There was a matrix $A$ $$A=\begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & ...
25
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2answers
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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 ?
25
votes
3answers
516 views

$\operatorname{tr}(AABABB) = \operatorname{tr}(AABBAB)$ for $2×2$ matrices

Similar to a previous question here, I wonder if cyclic permutations are the only relations amongst traces of (non-commutative) monomials. Since the evaluations $\operatorname{tr}:k\langle x,y,\dots ...
25
votes
7answers
2k views

How to tell if some power of my integer matrix is the identity?

Given an $n\times n$-matrix $A$ with integer entries, I would like to decide whether there is some $m\in\mathbb N$ such that $A^m$ is the identity matrix. I can solve this by regarding $A$ as a ...
25
votes
2answers
935 views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : ...
24
votes
6answers
1k views

Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
24
votes
1answer
1k views

Cauchy's Integral Formula for Cayley-Hamilton Theorem

I'm just working through Conway's book on complex analysis and I stumbled across this lovely exercise: Use Cauchy's Integral Formula to prove the Cayley-Hamilton Theorem: If $A$ is an $n \times n$ ...
24
votes
2answers
635 views

Countability of the zero set of a real polynomial

This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$? (By countable I mean countably ...
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 ...
23
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5answers
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Is there a definition of determinants that does not rely on how they are calculated?

In the few linear algebra texts I have read, the determinant is introduced in the following manner; "Here is a formula for what we call $detA$. Here are some other formulas. And finally, here are ...
23
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4answers
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How to intuitively understand eigenvalue and eigenvector?

I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman. Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to ...
23
votes
1answer
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$\sum_{k=0}^{100} a_k x^{k}=0, a_i \in \mathbb{Z}$ How many equations?

Let $x^{100}+a_{99} x^{99}+a_{98} x^{98}+ \dots +a_{1}x +a_{0}=0, \ \ a_{100}=1, a_i \in \mathbb{Z}$, is an algebraic equation with integer coefficients. Assume that all (100 with multiplicity) ...
22
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11answers
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Fun Linear Algebra Problems

I'm teaching a linear algebra course this term (using Lay's book) and would like some fun "challenge problems" to give the students. The problems that I am looking for should be be easy to state and ...
21
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4answers
2k views

Understanding Dot and Cross Product

What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them?
21
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4answers
695 views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
21
votes
1answer
513 views

How prove this matrix $\det (A)=\left(\frac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}\neq 0$

Question: let $a_{i}>1,i=1,2,3,\cdots,n$,and such $a_{i}\neq a_{j}$,for any $i\neq j$ define the matrix $$A=\left(\dfrac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}$$ show that: ...
21
votes
3answers
1k views

Natural and coordinate free definition for the Riemannian volume form?

In linear algebra and differential geometry, there are various structures which we calculate with in a basis or local coordinates, but which we would like to have a meaning which is basis independent ...
21
votes
1answer
253 views

Is a field determined by its family of general linear groups?

Assume that $K,L$ are fields such that there is an isomorphism of groups $\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$ for all $n \in \mathbb{N}$. Does it follow that $K \cong L$? I am also interested in ...
21
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0answers
874 views

Rank of a $n! \times n$ matrix

This question is about showing that $n!$ points resulting from applying a function (defined below) to the permutations of $n$ numbers lie on a $n-1$ dimensional hyperplane. Let $X=\langle ...