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Let U and W be subspaces of a nite dimensional vector space V such that U...

Let U and W be subspaces of a nite dimensional vector space V such that U ∩ W = {~0}. Dene their sum U + W := {u + w | u ∈ U, w ∈ W}.

(1) Prove that U + W is a subspace of V .

(2) Let U = {u1, . . . , ur} and W = {w1, . . . , ws} be bases of U and W respectively. Prove that U ∪ W is a spanning set for U + W.

(3) Prove that U ∪ W is linearly independent.

(4) Conclude that

dim(U + W) = dim U + dim W.

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