Definition A function f from vectors to vectors is linear if Such a function is called a linear transformation.

Theorem: Two linear transformations that agree on each member of some set of basis elements, agree everywhere.
In symbols:
If {bi} is a basis and for all i f(bi) = g(bi), and f and g are both linear then for all x f(x) = g(x).

Since {bi} is a basis, any x may be expressed as Σixibi for suitable choice of xi from the field. (Summation is over the number of dimensions.)
f(x) = f(Σixibi) = Σif(xibi) = Σixif(bi) = Σixig(bi) = Σig(xibi) = g(Σixibi) = g(x).