Let V be a vector space over R, and let Sị and S2 be subspaces of V. (a) Prove that Si C S2 = dim(S1) < dim(S2). (b) Prove that (Sı C S2 and dim(S1) = dim(S2)) → S1 = S2. (c) Prove that if u1,..., uk are linearly independent vectors in V, and v E V with v 4 span(u1,..., Uk), then u1, ..., Uk, v are linearly independent. %3D

Let V be a vector space over R, and let S₁ and S₂ be subspaces of V.

a)  Prove that S₁ ⊆ S₂ ⇒ dim(S₁) ≤ dim(S₂).

b)  Prove that (S1 ⊆ S2 and dim(S₁) = dim(S₂)) ⇒  S₁ = S₂

c)  Prove that if u1, . . . , uk are linearly independent vectors in V , and v v  not ∈ span(u1, . . . , uk), then u1, . . . , uk, v are linearly independent.