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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.
Add Linear algebra as a Learn subject and every problem in these areas routes to the Socratic tutor — attempt-first, hint-laddered, output-guarded.
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
the anti-pattern
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.
the recall card
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.
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
Attempt-first. Never the answer. Add it as a Learn subject in 30 seconds.