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Linear Transformations
Function Inverses
Linear Transformation Products
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Linear Isomorphisms
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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

\[ v_1 = T^{-1}(w_1) \quad \text{ and } \quad v_2 = T^{-1}(w_2) \]

In other words,

\[ Tv_1 = w_1 \quad \text{ and } \quad Tv_2 = w_2 \]

and so by the linearity of $T$,
Recall that we can use the terminology the inverse because inverses are unique (if they exist; see Function Inverses).


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 $ST$

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