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1. Let T: Mn(R) ➔ Mn (R) be a linear transformation from the vector space of n x n real matrices into itself such that T(A) = At . Find the minimal polynomial of T. 2. Let V be an n-dimensional vector spaces over F with basis {v1, · · ·, vn}, and let A= (O'.ij) be a fixed n x n matrix with Cl:ij E F. For any u = {1v1 + · · · + {nvn, v = 111v1 + · · · + 1/nVn, define a bilinear form on V x V as B(u, v) = u'::1 Lj=l ll:ij{i1/i· Show that B is nondegenerate if and only if A has rank n. 3. Let V, W be vectors spaces over F, Y be a subspace of V, and TE L(V, W). Show that [v] ➔ Tv, [v] E V/Y defines a linear transformation of V/Y ➔ W if and only if Y c n(T). Furthermore, if Y = n(T), the linear transformation is an isomorphism of V/Y onto the range T(V) of T. 4. Show that two n x n diagonal matrices with coefficients in F are similar if and only if the diagonal elements of one matrix are a rearrangement of the diagonal elements of the other one. 5. Let A E Cnxn be a unitary matrix and I+ A be invertible. Show that B = (I - A)(I + A)-1 is skew-Hermitian, that is, BH = -B. 6. Suppose A and Bare matrices in lower triangular form, show that AxB is also in lower triangular form. Furthermore, every eigenvalue of AxB has the form of afJ, where a is an eigenvalue of A and fJ is an eigenvalue of B.

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