Definition of invertible/nonsingular linear transformation
Created almost 9 years ago, updated 5 months ago
ID24021
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.
**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)