Linear Function as a Linear map

In linear algebra, a linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:

f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y})
f(a\mathbf{x}) = af(\mathbf{x}).

Here a denotes a constant belonging to some field K of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be K itself.

Some authors use “linear function” only for linear maps that take values in the scalar field;[4] these are also called linear functionals.

 

~source : http://en.wikipedia.org/wiki/Linear_function