Ok I am doing some additional problems becuase my professor assigns evens and i do odd and evens so i can at least check my work in backs of book. Prove that if W is a subspace of vector space V, dim(W) is less than or equal to dim(V). Proof: We know W is a subspace of V and therefore has a basis. The basis in W is in V and we also know V has a basis. If the basis for W has same dimension as a basis for V then W is V and we are done. Since V contains W that means the basis for W must be a linear combination of vectors in basis of V. So since every w in W can be generated by a basis in V then then dimV cannot be smaller than dimW. there for dimW<=dimV.