Definition: A linear transformation \(T: V \to W\) is invertible (or nonsingular) if there exists a linear transformation \(T^{-1}: W \to V\) such that:
\[T^{-1} \circ T = I_V \quad \text{and} \quad T \circ T^{-1} = I_W\]
The term nonsingular is sometimes used, particularly when the transformation is represented by a matrix: a matrix is nonsingular if and only if its determinant is nonzero.
Equivalent conditions (for finite-dimensional spaces):
- \(T\) is invertible
- \(T\) is one-to-one and onto
- \(\ker(T) = \{\mathbf{0}\}\)
- \(\text{range}(T) = W\)
- \(\dim(V) = \dim(W)\) and \(T\) is one-to-one
- The matrix of \(T\) is nonsingular (determinant is nonzero)
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