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 ...

learn more… | top users | synonyms (1)

351
votes
10answers
55k 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 ...
197
votes
4answers
84k 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 ...
150
votes
6answers
10k views

Why does this matrix give the derivative of a function?

I happened to stumble upon the following matrix: $$ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix} $$ And after trying a bunch of different examples, I noticed the ...
126
votes
1answer
6k views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
122
votes
20answers
25k 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 ...
116
votes
5answers
5k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
107
votes
8answers
71k views

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
93
votes
18answers
12k views

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 ...
88
votes
1answer
6k 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 ...
81
votes
23answers
12k views

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that non of the $a_k$'s is a linear ...
81
votes
5answers
11k 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 ...
71
votes
10answers
3k views

Why, intuitively, is the order reversed when taking the transpose of the product?

It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the following sentence: The opposite of putting ...
70
votes
3answers
3k 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: ...
70
votes
17answers
66k views

Where to start learning Linear Algebra? [closed]

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 ...
69
votes
4answers
11k 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?
59
votes
5answers
39k views

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 ...
49
votes
6answers
8k views

Why does the Cauchy-Schwarz Inequality even have a name?

When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of proof. I've always thought the geometric definition ...
42
votes
6answers
3k 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 ...
38
votes
11answers
68k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
21
votes
3answers
9k views

Geometric interpretation for complex eigenvectors of a 2x2 rotation matrix

The rotation matrix $$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and ...
17
votes
4answers
12k views

Is there a 3-dimensional “matrix” by “matrix” product?

Is it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form? I would appreciate it if you could help me to find out some tutorials online or mathematical ...
14
votes
1answer
4k views

Inverse of a Toeplitz Matrix

A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is an $n\times n$ Toeplitz matrix: $$ A ...
10
votes
1answer
12k views

Why are all nonzero eigenvalues of the skew-symmetric matrices pure imaginary?

Assume that $A$ is an $n\times n$ skew-symmetric real matrix, i.e. $$A^T=-A.$$ Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and ...
8
votes
9answers
57k views

Finding a unit vector perpendicular to another vector

For example we have the vector $8i + 4j - 6k$, how can we find a unit vector perpendicular to this vector?
8
votes
3answers
3k views

Why are Vandermonde matrices invertible?

A Vandermonde-matrix is a matrix of this form: $$\begin{pmatrix} x_0^0 & \cdots & x_0^n \\ \vdots & \ddots & \vdots \\ x_n^0 & \cdots & x_n^n \end{pmatrix} \in ...
7
votes
2answers
1k views

Advice: Modern vs. Classics

First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if ...
6
votes
2answers
29k views

Inverse matrix's eigenvalue?

It's from the book "linear algebra and its application" by gilbert strang, page 260. $(I-A)^{-1}$=$I+A+A^{2}+A^{3}$+... Nonnegative matrix A has the largest eigenvalue $\lambda_{1}$<1. Then, ...
6
votes
2answers
676 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

I'm searching for some material (books or lecture notes) that extensively uses a geometric approach to explain the meaning of the concepts realted regarding to vector spaces, matrices, and linear ...
6
votes
1answer
2k views

What's a good book on advanced linear algebra?

I'm taking an advanced linear algebra course and I'm a little confused about books. The teacher said we could use any book we wanted to, but he recomended just Hoffman and Kunze and also Kostrikin, ...
6
votes
0answers
56 views

In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
5
votes
2answers
7k views

How to show that a given set is a vector space?

I am having some issues with this problem in my Linear Algebra textbook. The goal is to either show that the given set, W, is a vector space, or to find a specific example to the contrary: ...
5
votes
1answer
62 views

Just being inquisitive..

This is a general question that came to my mind while listening to a lecture(although its framing may make it look like a textbook question). Suppose that $A$ and $B$ be real matrices. $A$ is ...
5
votes
1answer
154 views

Construction of Free Vector Spaces

I've been studying the construction of Free Vector Spaces and I want to confirm if my conclusions are correct. Given a set $A$ we wish to construct a vector space $F(A)$ which intuitively is the ...
4
votes
4answers
780 views

Exercise books in linear algebra and geometry

I'm studying Brannan's Geometry and Lang's Introduction to Linear Algebra and I was wondering if there are some exercise books (that is, books with solved problems and exercises) that I can use as ...
4
votes
3answers
164 views

Can an eigenvalue (of an $n$ by $n$ matrix A) with algebraic multiplicity $n$ have an eigenspace with fewer than $n$ dimensions?

Is it possible for a matrix with characteristic polynomial $(λ−a)^3$ to have an eigenline (one-dimensional eigenspace)? I know that geometric multiplicity can generally be smaller than algebraic ...
4
votes
1answer
68 views

Jordan normal form over $\mathbb{C}$

Let there be $T:\mathbb{C}^8\rightarrow \mathbb{C}^8$ Such that $ T\left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} ...
4
votes
2answers
43 views

The geometric action of an orthogonal $3 \times 3$ matrix with determinant $-1$.

I am trying to prove that the action of an orthogonal $3\times 3$ matrix with determinant $-1$ is a reflection about an eigenvector in one of the matrix's eigenspace, but I am a little lost. Problem ...
3
votes
3answers
110 views

Proof: Sum of dimension of orthogonal complement and vector subspace

Let $V$ be a finite dimensional real vector space with inner product $\langle \, , \rangle$ and let $W$ be a subspace of $V$. The orthogonal complement of $W$ is defined as $$ W^\perp= \left\{ v ...
3
votes
2answers
673 views

Sum of projections

Let $E_1$ and $E_2$ be projections on $V$, a vector space over $F$. Why is if $\operatorname{char}F\neq2$ then $E_1+E_2$ is a projection iff $E_1E_2=E_2E_1=0$ ?
3
votes
1answer
311 views

Given a reduced row exhelon form of a $4 \times 4$ matrix and two columns, how do you find the other two columns?

I am given the following : Let $A$ be a $4 \times 4$ matrix with RREF given by: $$ U = \begin{bmatrix} 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \\ 0 ...
3
votes
2answers
3k views

Pseudoinverse matrix and SVD

I'm trying to solve an homework question but I got stuck. Let A be a m x n matrix with the SVD $A = U \Sigma V^*$ and $A^+ = (A^* A)^{-1} A^*$ its pseudoinverse. I'm trying to get $A^+ = V ...
3
votes
2answers
21 views

Error Correcting Code

In my Linear Algebra book I have a chapter about error correcting code. there is an example involving Redundancy in the form of a check digit : we have $white=(0,0)$, $red=(1,0)$, $blue=(0,1)$, ...
3
votes
1answer
99 views

Why is the nuclear norm called so?

A simple question. Why is the sum of the singular values of a matrix called its nuclear norm? What is the origin of, and motivation for, this term? Apparently the term nucleus is sometimes used to ...
3
votes
1answer
242 views

Problem with sum of projections

Let $X$ be a real linear space, $(P_i)_{i=1}^n$ -a finite sequence of linear mappings $P_i :X\rightarrow X$ such that $P_i^2=P_i$ for $i=1,...,n$, $(P_1+...+P_n)^2=P_1+...+P_n$. I wish to show ...
3
votes
0answers
56 views
+50

Linear algebra, are my steps to compute cost function value correct?

Reading this : From http://www.holehouse.org/mlclass/09_Neural_Networks_Learning.html Cost function for a single training example is given as : cost(i) = $ y^i \; log \; h_\theta(x^i) + (1 - ...
3
votes
1answer
55 views

Detecting singular system during Cholesky resolution

I am solving small linear systems with a symmetric positive matrix by the method of Cholesky, without pivoting. "Bad" matrices are detected when you take the square root of a diagonal element, which ...
3
votes
0answers
125 views
+50

Spliting subspaces and finite fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
2
votes
1answer
56 views

$A$ and $B$ are positive matrices and $\|A+B\| = \|A\|+\|B\|$, then A and B have a common eigenvector

Suppose $A$ and $B$ are positive matrices such that $\|A+B\| = \|A\|+\|B\|$. Show that $A$ and $B$ have a common eigenvector, where $\|A\|$ is the operator norm of $A$. I'm wondering if there is a ...
2
votes
2answers
16 views

Elementary tensors

Let $G,H$ be $R$-modules, and $G \otimes H$ be it's tensor product. I can't prove it and I suspect it's false that any element $\tau \in G \otimes H$ can be written as $\tau = g \otimes h$ for some ...
2
votes
2answers
39 views

What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...