Q2 part a to e parts..need solution in neat and clear handwriting on A4 page ASAP

THEOREM 4.2.1 If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the following conditions are satisfied. (a) If u and v are vectors in W, then u + v is in W. (b) If k is a scalar and u is a vector in W, then ku is in W. Proof If W is a subspace of V, then all the vector space axioms hold in W, including Axioms 1 and 6, which are precisely conditions (a) and (b). Conversely, assume that conditions (a) and (b) hold. Since these are Axioms 1 and Theorem 4.2.1 states that W is a subspace of V if and only if 6, and since Axioms 2, 3, 7, 8, 9, and 10 are inherited from V, we only need to show it is closed under addition and that Axioms 4 and 5 hold in W. For this purpose, let u be any vector in W. It follows scalar multiplication. from condition (b) that ku is a vector in W for every scalar k. In particular, Ou = 0 and (-1)u = -u are in W, which shows that Axioms 4 and 5 hold in W. 2. Use Theorem 4.2.1 to determine which of the following are subspaces of My. (a) The set of all diagonal n x n matrices (b) The set of all n x n matrices A such that det ( A) = (. (c) The set of all n x n matrices A such that tr(A) = 0. (d) The set of all symmetric n x n matrices. (c) The set of all n x n matrices A such that AT = -A. (f) The set of all n x n matrices A for which Ax = 0 has only the trivial solution. (g) The set of all n x n matrices A such that AB = BA for some fixed n x n matrix B