q 10 Suppose V is ferrite dimensional and U is a subspace of

q 10

Suppose V is ferrite dimensional and U is a subspace of V. Prove that there exists a subspace W of V such that V = U + W and U W - {0}, where 0 is the additive identity of V. Let U_1 and U_2 are two subspaces of a finite dimensional vector space V Then prove that.

Solution

Let V = span {v₁,v₂,…,v_n} and U= span(u₁,u₂,…u_m} where m n. Also, let W= U^c = span { w₁, w₂,…w_n-m}.

Then W is a subspace of V and U +W = V.