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)).
**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 \(T^{-1}(u,v) = (u/2, v/3)\).