Definition of invertible/nonsingular linear transformation
Definition: A linear transformation (T: V o W) is invertible (or nonsingular) if there exists a linear transformation (T^{-1}: W o V) such that:
[T^{-1} \circ T = I_V \quad ext{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}})
- ( ext{range}(T) = W)
- (\dim(V) = \dim(W)) and (T) is one-to-one
- The matrix of (T) is nonsingular (determinant is nonzero)