Let L:V --> be an isomorphism of vector space V onto vector space W. Show that L(v-w)= L(v) - L(w)

If v = [a1, a2, a3] and w = [b1, b2, b3]

L(v - w) = L(a1 - b1, a2 - b2, a3 - b3)

How do I take the next step in proving that L(v) - L(w)? It seems too simple to just separate it into L(a1, a2, a3) - L(b1, b2, b3) which would then equal L(v) - L(w). Am I missing a step, or is it that simple? Thanks!