Definition of inverse of a linear transformation

**Definition:** If \(T: V \to W\) is an invertible linear transformation, its *inverse* is the unique linear transformation \(T^{-1}: W \to V\) satisfying:
\[T^{-1}(T(\mathbf{v})) = \mathbf{v} \quad \text{for all } \mathbf{v} \in V\]
\[T(T^{-1}(\mathbf{w})) = \mathbf{w} \quad \text{for all } \mathbf{w} \in W\]

**Properties of the inverse:**
- \(T^{-1}\) is also linear
- \((T^{-1})^{-1} = T\)
- If \(T\) and \(S\) are both invertible, then \((ST)^{-1} = T^{-1}S^{-1}\)
- If \(T\) is represented by matrix \(A\), then \(T^{-1}\) is represented by \(A^{-1}\)

**Example:** If \(T(x,y) = (2x, 3y)\), then \(T^{-1}(u,v) = (u/2, v/3)\).Copy