• f(a+b) = f(a) + f(b) for any vectors, a and b,
• k*f(a) = f(k*a) for any field element k and any vector a.

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

Proof:
Since {b_i} is a basis, any x may be expressed as Σ_ix_ib_i for suitable choice of x_i from the field. (Summation is over the number of dimensions.)
f(x) = f(Σ_ix_ib_i) = Σ_if(x_ib_i) = Σ_ix_if(b_i) = Σ_ix_ig(b_i) = Σ_ig(x_ib_i) = g(Σ_ix_ib_i) = g(x).
Q.E.D.