Question

9. Let S and T be two subspaces of some vector space V.

(b) Define S + T to be the subset of V whose elements have the form (an element of S) + (an element of T). Prove that S + T is a subspace of V.

(c) Suppose {v₁, . . . , v_i} is a basis for the intersection S ∩ T. Extend this with {s₁, . . . , s_j} to a basis for S, and separately with {t₁, . . . , t_k} to a basis for T. So we have that the {v₁, . . . , v_i , s₁, . . . , s_j} is a basis for S, and {v₁, . . . , v_i , t₁, . . . , t_k} is a basis for T. Prove that {v₁, . . . , v_i , s₁, . . . , s_j , t₁, . . . t_k} is a linearly independent set.

Note: dim S + dim T = dim (S ∩ T) + dim (S + T) (i.e., (i + j) + (i + k) = i + (i + j + k)).

(d) (1 point) Suppose V is M_3×3, S is the subspace of all symmetric 3 × 3 matrices, and T is the subspace of all upper triangular 3 × 3 matrices. Show by example that S ∪ T not a subspace of V.

Answer 1