In this question, we'll figure out how many subspaces a real or complex vector space can have.

So let V be a real or complex vector space of dimension 2 or greater. Suppose we have two vectors {v, w} in V which are linearly independent.

A] Show that for any nonzero distinct scalars α and β, the pair of vectors {(v + αw), (v + βw)} is also linearly independent.

B] From the previous part, explain why span({v + αw}) ̸= span({v + βw})

C] Conclude that V has infinitely many distinct subspaces.

D] If now dim(V ) = 1, how many distinct nonempty subspaces does V have?

E] How many distinct nonempty subspaces does ⟨⃗0⟩ have?