University of California – Irvine Algebra Mathematics Questions

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2. Let F be a field and let V be a finite dimensional vector space over F.
(a) (3 points) Let End(V) be the set of F-linear transformations : V + V. For 0,4 E
End(V), we define 0+¥ to be the function V + V that sends v to (v) +¥(v). Prove
that 0 + 4 E End(V).
(b) (4 points) Prove that End(V) is a ring, where addition is as in the previous part and
multiplication is composition.
(c) (4 points) Prove that End(V) is commutative if and only if V has dimension 1.