Linearity

Preserving vector operations:

• scalar multiplication: $f(ax) = ax$, and
• addition: $f(x + y) = f(x) + f(y)$.

A linear operator then is determined by its values on the elements of the basis of the domain space, as they are expressed as linear combinations of the elements of the basis of the target space. The coefficients of these linear combinations form the columns of the matrix of linear operator.