- 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_{i}x_{i}b_{i} for suitable choice of x_{i} from the field. (Summation is over the number of dimensions.)

f(x) = f(Σ_{i}x_{i}b_{i})
= Σ_{i}f(x_{i}b_{i})
= Σ_{i}x_{i}f(b_{i})
= Σ_{i}x_{i}g(b_{i})
= Σ_{i}g(x_{i}b_{i})
= g(Σ_{i}x_{i}b_{i})
= g(x).

Q.E.D.