Looking ahead: encode all sequential computation pipelines

Numerical pipelines in graphics, robotics, and recurrent nets are long matrix chains. This chapter is the sign-convention bedrock .

Associativity allows grouping long products without changing the answer, even when matrices do not commute. Inverse of $AB$ for invertible square factors is $B^{-1}A^{-1}$: reverse order when inverting composition.

Check dimensions at every stage; a mismatch is cheaper to catch before animation than after. Robotics forward kinematics is one pipeline where applying joint transforms left-to-right must mirror the physical sequence of joints.

Keep the matrix nearest the vector as the first map applied. That convention aligns function composition $f(g(x))$ with $FG\mathbf{x}$ when column vectors are used consistently .

Check your understanding. The tasks below rest on these ideas: Correct: associativity (free bracketing) is independent of commutativity (free reordering). Not quite: regrouping needs no commuting or squareness, and it never changes the answer, so 'never' is wrong. Correct: to undo 'apply $B$ then $A$', you undo $A$ first, then $B$: $B^{-1}A^{-1}$. Not quite: keeping the order, reusing $AB$, or transposing does not invert the product. Correct: with the column-vector convention, the inner map $g$ sits next to $\mathbf{x}$ just as $g$ is applied first inside $f(g(x))$. Not quite: the correspondence needs linearity, not affineness, and has nothing to do with orthogonality or inverses. Correct: chained joint transforms compose like matrix products, so reversing the order mirrors the wrong physical motion. Not quite: pixel addition, sorting, and a lone dot product involve no composition order to get wrong.