Start here: linear algebra
These guides are a companion to learning linear algebra, not a whole course. This page covers what you should know first, what to read alongside, and which guide to open when.
Are you ready?
Try each of these before opening the answer. The first three are all you need for the field guide and the questions guide. The fourth is for the problems page.
answer
Add the equations: $3x = 6$, so $x = 2$ and $y = 1$.
If that felt shaky, spend a week on school algebra first. Linear algebra is this, many times over.
answer
$(1, 5)$: start at $(1, 2)$ and step up three times. If you haven't met vectors at all, the first two videos in the 3Blue1Brown series below cover everything you need.
answer
$f(g(2)) = f(3) = 9$, but $g(f(2)) = g(6) = 7$. Doing one thing and then another depends on the order. That is exactly why matrix multiplication doesn't commute.
answer
Write them as $2m$ and $2n$. Their sum is $2(m + n)$, which is even.
If you couldn't write something like that, read the first few chapters of a proof book before the problems page. Richard Hammack's Book of Proof is free online. Then come back. The problems page will still be here.
Which guide to open
- Brand new
Watch the first few 3Blue1Brown videos, then play with the field guide until the plates make sense. Then pick a textbook and work through the problems alongside it, one part at a time.
- Took a course, still shaky
Go to what everyone asks and jump to the chapter you're stuck on. Most people find their confusion is someone else's most-viewed question.
- Want it to stick
Do the problems without a book. Use the hints freely and the solution sketches sparingly.
What to read alongside
- 3Blue1Brown, Essence of linear algebra. Free videos, and the best first contact there is. Pictures before formulas.
- Gilbert Strang, Introduction to Linear Algebra, with MIT 18.06. The lectures are free. Computational and friendly, the course most engineers take.
- Sheldon Axler, Linear Algebra Done Right. Free online since the 4th edition. Proof-based and clean. The problems page follows a similar route.
- Jim Hefferon, Linear Algebra. Free, with worked answers to its exercises. Good if you want more problems.
How the pieces line up
Each part of the problems page, with where to read about it and where it turns up in the other two guides.
| problems | 3Blue1Brown | Axler (4th ed.) | questions guide | field guide |
|---|---|---|---|---|
| Spaces | chapters 1–2 | chapters 1–2 | span and bases | not covered |
| Maps | chapters 3, 4, 7, 8 | chapter 3 | matrices aren't numbers, kernel and image | every plate: each matrix is a map |
| Volume | chapter 6 | chapter 9 | the determinant | the flipped F, and the collapsers |
| Eigenvectors | chapters 13–14 | chapter 5 | eigenvectors | stretchers, turners and sliders |
| Perpendicular | chapter 9 | chapters 6–7 | special matrices, data | symmetric stretch, rotation |
Axler leaves determinants until the end on purpose, so if you're reading along, do the Volume part last.