Question

9. Let S and T be two subspaces of some vector space V. (b) Define S...

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, . . . , vi} is a basis for the intersection S ∩ T. Extend this with {s1, . . . , sj} to a basis for S, and separately with {t1, . . . , tk} to a basis for T. So we have that the {v1, . . . , vi , s1, . . . , sj} is a basis for S, and {v1, . . . , vi , t1, . . . , tk} is a basis for T. Prove that {v1, . . . , vi , s1, . . . , sj , t1, . . . tk} is a linearly independent set.

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

(d) (1 point) Suppose V is M3×3, 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 ST not a subspace of V.

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