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Definition of invertible linear transformation

Created over 8 years ago, updated 10 days ago

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).