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Linear Algebra

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Matrix multiplication

Phase I – II · Foundations to maps

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  1. Vectors and spacesRevised
  2. Linear combinationsCovered
  3. Span and independenceCovered
  4. Linear mapsCovered
  5. Matrix multiplicationLearning
  6. Determinants
  7. Inverses

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Phase II · Transformations

Matrix multiplication

Two moves, one after the other

A matrix moves every vector somewhere. Multiplying AB means: do B first, then A. That is the whole reason the order matters.

  • Rows of A meet columns of B — multiply pairwise, add.
  • (2 × 3) · (3 × 2) gives a (2 × 2). The inner sizes must match.
  • AB ≠ BA in general. Rotating then stretching isn’t stretching then rotating.
Composition, not arithmetic — that’s the idea I kept missing.
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