Linear Algebra question. I need Original solution. Thank you.

(10 pts) Let H and K be subspaces of a finite dimensional vector space V. Parts (a) and (b) below will will prove the following statement: If HQ)K = 0 (meaning that the intersection of the two spaces only contains the 0 vector), then dim(H + K) - dim(H) + dim(K). (btw - you don't need to do number 1 to do this problem; For this problem We are assuming H-+K is a subspace of V) (a) Suppose dim(H) = n, and dim(K) = m. Let {h1, ..h, } be a basis for H and {ki, ...km) be a basis for K. Show that {h1, ...h,, ki, ...km} spans H+K. (b) Show that {h1, ..ha, ki, ...km) is linearly independent (hint: In your linear independence equation set to 0, bring all the k terms to the right hand side and then use the assumption that H( K = 0). Then explain why therefore that dim(H + K) = dim(H) + dim(K)