Problem Set 11 (due November 29)MATH 110: Linear AlgebraEach problem is worth 10 points.PART 11. Curtis p. 276 2.2. Curtis p. 276 4.3. Curtis p. 276 9.4. Curtis p. 288 4.PART 2Problem 1(20)Show that for any 1 × n row matrix X, and any n × n matrix A, we havethatXAXt= XBXtwhere B is the symmetric matrix B =12(A + At).Problem 2(10)Let T : V → V be a symmetric transformation on a real Euclidean spaceV with quadratic form Q(x) = (T (x), x). Prove that if Q does not changesign sign on V (i.e. Q(x) ≥ 0 ∀x ∈ V or Q(x) ≤ 0 ∀x ∈ V ) then Q(x) = 0implies that T (x) = 0. (Hint: consider the function p(t) = Q(x + ty) andshow that p0(0) = 0).Problem 3 (10)a) Prove that the set of symmetric matrices forms a subspace of dimensionn(n+1)2in the vector space of real n × n matrices (which has dimension n2).b) Prove that the set of skew-symmetric matrices forms a subspace of di-mensionn(n−1)2in the vector space of real n×n matrices (which has dimensionn2).Problem 4 (10)Let T be a symmetric linear transformation on a real Euclidean space.Prove that T has a “square root”, i.e. there is a symmetric linear transfor-mation S such thatT = S2.Problem 5 (10)Define the index of a real symmetric matrix A to be the number of strictlypositive eigenvalues of A minus the number of strictly negative eigenvalues1of A. Let QA(x) and QB(x) be the quadratic forms associated with A andB, and suppose thatQA(x) ≤ QB(x)for all vectors x. Prove that the index of A is less than or equal to the indexof B.Optional ProblemLet T : V → V be a symmetric linear transformation on a real Euclideanspace V and let Q(x) = (T (x), x). Assume that there is an extremum (max-imum or minimum) at u for Q(x) among all the values that Q takes on theunit sphere (i.e. Q(u) is either a max or a min amongst all the values Q(x)with (x, x) = 1). Show that u is an eigenvector for T
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