Composition, Matrix Products, and Inverse Transformations

Modern interactive lesson

See matrix multiplication as a chain of transformations. Compare orders, compose rotations, and test inverse maps.

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Big idea: multiplication means composition

If $T(x)=Bx$ and $S(y)=Ay$, then applying $T$ first and $S$ second gives

$$(S\circ T)(x)=A(Bx)=(AB)x.$$

Read products from right to left: in $ABx$, $B$ acts first and $A$ acts second.

First apply $B$, then apply $A$.

First apply $A$, then apply $B$.

Usually $AB\ne BA$. Geometry makes this visible: rotating then projecting is different from projecting then rotating.

noncommutativity

Choose two transformations. The left panel shows $AB$ and the right panel shows $BA$ on the same input vector.

$A$ acts second in $AB$

$B$ acts first in $AB$

input vector $x$

A11

A12

A21

A22

B11

B12

B21

B22

angles add

angle $\theta$ degrees

angle $\phi$ degrees

An inverse transformation reverses a linear transformation. If $A$ is invertible, then $A^{-1}Ax=x$.

reverse map

vector x

Important: a projection has no inverse because many input vectors collapse to the same output.

$S\circ T$ corresponds to $AB$ when $T(x)=Bx$ and $S(y)=Ay$.

Rightmost transformation acts first.

$(AB)^{-1}=B^{-1}A^{-1}$.

In $PQx$, which transformation acts first?