Linear algebra
Vectors are directions, not numbers.
the shape of this subject
Linear algebra clicks the moment you stop seeing matrices as grids of numbers and start seeing them as transformations. Brainback pushes for that shift with one guiding question at a time.
Because linear algebra rewards the students who do the reps — and punishes the ones who outsource them.
- discipline
- STEM
- parent subject
- Mathematics
- typical weekly load
- 3-6 problem sets · 3-5 hours · 1-2 recall sessions
- covered on plan
- all plans (free forever)
Every linear algebra concept Brainback covers.
Add Linear algebra as a Learn subject and every problem in these areas routes to the Socratic tutor — attempt-first, hint-laddered, output-guarded.
What one turn in linear algebra actually looks like.
Attempt → correction → guiding question. No answer. Ever. Verbatim from a beta session.
I row-reduced to echelon and got a pivot in column 3…
What does this matrix do to the standard basis vectors?
no answer · one guiding question · your next move
How real attempts start. How Brainback nudges.
- I row-reduced to echelon and got a pivot in column 3…
- What does this matrix do to the standard basis vectors?
the anti-pattern
What linear algebra looks like when the AI does the thinking.
Answer machines make linear algebra look like a solved problem. It isn't. The exam room reveals whose reasoning was built and whose was rented.
- M1Applying a formula without checking whether the conditions actually hold.
- M2Skipping the sanity check — 'does the sign / magnitude make sense?'
- M3Memorizing steps instead of reasoning about why they're legal.
- M4Reading past the setup and diving straight into computation.
the recall card
What a spaced-repetition card looks like in linear algebra.
Every wrong attempt and low-scored explain-back becomes a card. SM-2 schedules them — the tricky ones come back tomorrow, the solid ones drift out to next week, then next month.
Recall the core concept in your own words. Then apply it to a fresh example under 2 minutes.
Related subjects.
The subject where the shortcut has always been most expensive.
Limits, rates, and shapes of change — earned, not delivered.
Where symbolic manipulation becomes second nature — or doesn't.
See the diagram. Not just look at it.
Unit circle, but also unit intuition.
The subject where wrong feels right — until Brainback slows you down.
linear algebra — ready when you are
Start your first linear algebra session.
Attempt-first. Never the answer. Add it as a Learn subject in 30 seconds.