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Problem 5. (Exercises 1 and 2 from Lecture 3.2) Let V be a vector space and let U, W C V be subspaces. (a) Finish the proof of the Second Isomorphism Theorem from class by showing that UnW = ker(f) where f is the linear transformation W -> (U + W)/U sending ww+ U. (b) Use the Second Isomorphism Theorem to show that if U and W are finite dimensional, then dim(U + W) - dim U + dim W - dim(Un W). Note: this is the "dimension version" of the Inclusion-Exclusion Formula from set theory: [XUY| =[X|+[Y|-[XnY| where | . | denotes cardinality. (c) Now assume U C W. Prove the Third Isomorphism Theorem: (V/U)/(W/U) ~ V/W. Hint: start by constructing a map V/U -> V/W