What everyone asks about linear algebra
This guide is built from the questions thousands of people have actually asked about linear algebra, in roughly the order they get stuck. Find the one that's bothering you, look at the real questions, then read the idea underneath them.
It assumes you've met the basics. If you haven't, start here first. How I built it from Stack Exchange data is at the end.
A short notation corner
About 5% of all the views go to questions that are really about a symbol. One of the biggest isn't even about linear algebra: 277,000 people have landed on a question about what $\nabla$ means. It's the gradient, from calculus.
| $\|v\|$ | length (norm) of $v$. Usually $\sqrt{v \cdot v}$. |
| $\langle u, v\rangle$ | inner product. On $\mathbb R^n$ it's the dot product; the brackets are for when it isn't. |
| $f : \mathbb R^n \to \mathbb R^m$ | a function eating length-$n$ vectors and returning length-$m$ ones. As a matrix it's $m \times n$, the order people get backwards. |
| $U \oplus W$ | a sum of subspaces that only meet at $0$, so every vector splits into its two parts in exactly one way. |
| $\ker A$, null space | the same thing. One name comes from algebra, the other from matrices. |
| trivial solution | $x = 0$, which always solves $Ax = 0$. "Non-trivial" means any other one. |
Where these questions come from
I pulled the 2,000 most upvoted linear algebra questions from Math Stack Exchange to see where people actually get stuck. They've been viewed 52 million times between them. The first thing that jumped out wasn't a topic. It was the grammar.
Questions that start with how do I, find, determine or show get the traffic. Questions that ask why, or ask for intuition or meaning, get the votes. Each dot below is one question.
Rank the questions by views and then by votes, and the two top tens barely overlap:
Most viewed
Most upvoted
Across the top hundred by views, 34 are how questions and 13 are why questions. Flip to the top hundred by votes and it's 12 and 33. A why question gets about 3 times as many upvotes per view, and about half as many views.
I read that as two different people. One has a problem set due tomorrow and needs the procedure. The other has done the procedure fifty times and has just realised they don't know what it means. Most people are the first person first and the second person later, if ever.