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)).