Invertible Linear Transformations
Motivating result
Suppose $T: V \to W$ is linear and $T^{-1}$
exists.
Then $T^{-1}$ is linear.
We show that $T^{-1}$ is additive and
homogenous.
Let $w_1, w_2 \in W$ and define $v_1$ and
$v_2$ so that
Recall that we can use the terminology
the inverse because inverses are unique (if
they exist; see Function Inverses).
\[ v_1 = T^{-1}(w_1) \quad \text{ and } \quad v_2 = T^{-1}(w_2) \]
In other words,\[ Tv_1 = w_1 \quad \text{ and } \quad Tv_2 = w_2 \]
and so by the linearity of $T$,Definition
A linear map $T \in \mathcal{L} (V, W)$ is invertible if there is a linear map $S \in \mathcal{L} (W, v)$ so that $ST$