composition of transformations
Composition just means doing one transformation and then feeding its output into another: first rotate, then stretch. The combined effect is itself a single linear transformation, so a whole chain of steps can be bundled into one map.
And here is the punchline of the entire subject: composing transformations corresponds exactly to multiplying their matrices. If S has matrix A and T has matrix B, then doing T first and then S has matrix A*B. The order reads right to left, matching how the vector flows through B before reaching A. This is the real reason matrix multiplication is defined the way it is, with its peculiar row-times-column rule.
It also explains why matrix multiplication is usually not commutative: A*B and B*A need not agree. Rotating then reflecting is generally not the same as reflecting then rotating, and the matrices honestly record that difference in order.
Chaining two maps equals multiplying their matrices; swapping the order usually changes the result.
Mind the order. The matrix written rightmost acts on the vector first. So A*B means: apply B, then A. It feels backward at first, but it lines up with how functions are written.