A function \(L\from \RNrSpc{n}\to \RNrSpc{m}\) is called linear if it has the two properties below

• \(L(\Vect{x}+\Vect{y}) = L(\Vect{x}) + L(\Vect{y})\) for all \(\Vect{x},\Vect{y}\in\RNrSpc{n}\)
• \(L(t\cdot \Vect{x}) = t\cdot L(\Vect{x})\) for all \(t\in\RNr\), and all \(\Vect{x}\in\RNrSpc{n}\).

Alternate terms for ‘linear function’ include: linear map, linear transformation, homomorphism (of vector spaces).