Composing Transformations

Linear Algebra (Matrices) · Axiom Academy

LESSON Composing Transformations Applying one linear map after another is a single matrix product BA — and that one fact is the reason matrix multiplication is defined the way it is. Apply T (a rotation, matrix A ) to a shape, then apply S (a shear, matrix B ). That is the composition . The single matrix BA sends the original shape to the same place in one step — watch the two-stage path on the left land exactly where the one-shot BA lands on the right. Do T first, then S — the matrix of the composition is the product BA Track what happens to a vector : . Associativity lets us collapse into — so the combined matrix has to be BA . The animation builds BA the only way that makes this work: each entry is a row of B dotted with a column of A . A acts on first, so A sits on the right, next to . B acts on the result, so B sits on the left — the map applied last is written first. Entry (i,j) of BA is row i of B times column j of A — that rule is forced, not arbitrary. for every , so the two maps fuse into the single matrix BA . Matrix multiplication is defined as "row times column" precisely so that equals . The composition came first; the multiplication rule was reverse-engineered to match it. Swapping the order swaps which map goes first — and that lands the shape somewhere else. On the left we do (rotate, then shear), matrix BA ; on the right we do (shear, then rotate), matrix AB . Same two maps, same starting shape, different result. So matrix multiplication is not commutative .

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