NAME:1 /30 2 /10 3 /12 4 /18 5 /10 6 /8 7 /12 T /100MATH 430 (Fall 2005) Exam 2, November 3rdShow all work and give complete explanations for all your answers.This is a 75 minute exam. It is worth a total of 100 points.(1) [30 pts](a) Let u be a non-zero n × 1 column vector and v a non-zero m × 1 column vector. Prove that uvThasrank 1.(b) Suppose that A and B are n × n matrices. Prove that trace(AB) = trace(BA).(c) Let T : V → V, be a linear operator, where V is a finite dimensional vector space. Using (b), definetrace(T).(d) State the three properties that characterize the determinant as a function from the space of n × n realmatrices to R.2(e) Suppose that A and B are n × n invertible matrices. Using the definition you gave in (d) to prove thatdet(AB) = det(A) det(B).(f) Let u be a length one vector in Rn, and let R be the n × n matrix R = In− 2uuT. Calculate det(R),and explain the physical meaning of the linear operator defined by R(v) = Rv.3(2) [10 pts] True or false? If true give a brief justification. If false provide a counterexample.(a) det(A + B) det(A − B) = det(A2− B2).(b) Let v = (2, 3)T. In the standard basis B for R2, the matrix of the projection operator Pv: R2→ R2onto the span of v is[Pv]B=4 66 9.4(3) [12 pts] For the linear operator T : R2→ R2defined by T(x, y) = (x − y, 2x + 4y), calculate thematrix, [T]B, of T in the basis B =11,21.5(4) [18 pts] Let P be the matrixP =1 2 30 6 67 8 9.(a) Calculate det(P) using(i) Row operations(ii) Block determinants based on the blockingP =A BC D, where A is 1 × 1 and D is 2 × 2.6(iii) A cofactor expansion.(b) What is det(PTP), and why?7(5) [10 pts] Let T : V → W be a linear transformation between finite-dimensional vector spaces V and W.Let B be a basis for V and let B0be a basis for W. Define the matrix [T]BB0of T with respect to thesetwo bases, and prove that[T(u)]B0= [T]BB0[u]B.8(6) [8 pts] Suppose A is a square matrix whose entries are differentiable functions of a real variable t, thatis, Aij= Aij(t). Prove that det A is also a differentiable function of t.9(7) [12 pts] The least squares quadratic fit to m data points (t1, y1), (t2, y2), · · · (tm, ym) in R2is thequadratic function y = f(t) = α + βt + γt2for which the parameter vector (α, β, γ) is the global minimumof the functionQ = Q(α, β, γ) =mXi=1(α + βti+ γt2i− yi)2.(a) Let x = (α, β, γ)T. Find an m × 1 vector y and an m × 3 matrix A so thatQ = Q(x) = ||Ax − y||2.10(b) By differentiating Q(x) with respect to the i-th coordinate xiof x, prove that the minimizer of Qsatifies the normal equations ATAx = ATy.Pledge: I have neither given nor received aid on this