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Q. V is a finite dim. vsp over K, and W is a subspace of V. Let f be a linear functional on W. Show that there exists a linear functional g on V s. t. g(w)=f(w).

Ans. So far:

For all w in W, w = c_1*v_1 + c_2*v_2 + ... + c_n*v_n (where v_i for all i is the basis of V and c_i for all i are scalars part of K).

f(p*w1 + q*w1) = p*f(w1) + q*f(w2)

But since w for any w can be represented by a linear comnbination of the basis for V, there exists g, s.t.

f(w) = c_1*g(v_1)+c_2*g(v_2)+....

How can I complete this? Can I make the above assertion?