Definition: A linear transformation \(T: V \to W\) is invertible if there exists a linear transformation \(S: W \to V\) such that:
\[S \circ T = I_V \quad \text{and} \quad T \circ S = I_W\]
where \(I_V\) and \(I_W\) are the identity transformations on \(V\) and \(W\) respectively.
For finite-dimensional spaces, the following are equivalent:
- \(T\) is invertible
- \(T\) is one-to-one and onto
- \(\ker(T) = \{\mathbf{0}\}\) and \(\text{range}(T) = W\)
- \(\dim(V) = \dim(W)\) and \(T\) is one-to-one (or onto)
- The matrix of \(T\) (relative to any bases) is nonsingular (has nonzero determinant)
Example: Rotation by any angle \(\theta\) in \(\mathbb{R}^2\) is invertible; its inverse is rotation by \(-\theta\).
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