A linear function \(L\from \RNrSpc{n}\to \RNrSpc{m}\) satisfies \(L(\Vect{0})=\Vect{0}\).
A linear function \(L\from \RNrSpc{n}\to \RNrSpc{m}\) satisfies \(L(\Vect{0})=\Vect{0}\).
The proof are about to give might appear silly at first sight because we appear to perform pointless manipulations of zero vectors. However, these manipulations of zero vectors are designed to relate to the two fundamental properties which characterize linear functions
| \(L(\Vect{0})\) | \(=\) | \(L(\Vect{0} + \Vect{0})\) |
| \(\) | \(= \) | \(L(\Vect{0}) + L(\Vect{0})\) |
| \(\Vect{0}\) | \(=\) | \(L(\Vect{0})\) |
This was to be shown.