2. Inverses of homogeneous linear transformations

In the preceding section, $T$ and $U$ are inverses of each other if $T \circ U$ and $U \circ T$ are both identity functions, i.e., $T(U($x$)) =$   x, $U(T($x$)) =$   x. If so, write $U = T^{-1}$. This can be the case only if $m = n = k$, so that $T,U:$   R$^ n \rightarrow$   R$^n$.

For the corresponding matrices $A$ and $B$, this means that they must both be square of the same size and $AB = I$, $BA = I$. In other words, for $T = T _ A$ and $U = T _ B$, $U = T^{-1}$ if and only if $B = A^{-1}$.

Recall that one practical way to find the inverse of a matrix $A$ is to make a matrix $[A\vert I]$ and row-reduce it to get $[I\vert B]$; then $B = A^{-1}$. (For an orthogonal matrix, though, you'll see that it's much easier.)