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1. Let a1 = (x1,y1) and a2 = be two points in the plane R². Prove that the area of the parallelogram spanned by the origin and the two points is given by the absolute value of the determinant (D(a1,a2)1. 2. Let S = {1 n}. A permutation on n letters is a bijection o : S S. A permutation o is called a transposition if there exist i,j E {1,..., n} such that o(i) = j,o(j) = i and o (k) = k for all other k (i.e. o swaps i and j and keeps the remaining elements the same). (a) Given a permutation o, let T R" R" be the linear transformation permuting the standard basis, i.e. T(ei) = Prove that det(T) = 1. Define sgn(o), the sign of o as the value of this deter- minant. (b) Prove that every permutation can be expressed as a composition of transpoosi- tions. Moreover, prove that for a permutation o, sgn(a) = 1 if and only if it can be expressed as a composition of an even number of transpositions. Deduce that a permutation cannot be expressed as both a product of an even number of transpositions and an odd number of transpositions. (c) Let Sn denote the set of all permutations on n letters. Let A = (aij) be an n by n matrix. Prove that = 3. Let x1 xn E R be an n-tuple of numbers. Prove that = (xj - ii). Kikjen 1 In (Hint: Let C1 Cn be the columns of the matrix; use the fact that D(C1 cn) = - Then take row expansion along the first row and use induction.) 4. Let T : V V be a linear transformation and let U1 Un be a basis of V consisting of eigenvectors of T with eigenvalues X1 In, respectively; i.e. T(vi) = livi. (a) Prove that f(T) = 0, where f(x) = (x - (1) (x - 12) (x-An). (b) Prove that the minimual polynomial of T is II(x-Ai), where the li are the distinct eigenvalues of T. (c) Show that the matrices 1 0 0 0 2 0 -1 0 0 2 and 0 0 2 0 2 - 1 - 1 have the same minimal polynomials. 5. Let a(x), b(x) E F(x) be two nonzero polynomials. Apply the division process to obtain a = bqo + TO b = roqi + T1, deg(ri) < degro To = T192 + T2, deg(r2) < degri Ti = Ti+1Qi+2 + Ti+2, deg(ri+2) < deg(ri+1). Prove that for some 10, Tio # 0 and Tio+1 = 0 and prove that Tio is the greatest common divisor of a and b. 6. For each of the following, find the greatest common divisor of the given polynomials in R[x]: (a) 4.3 + 2x2 - 2x - 1, 2x³ - x2 + x + 1 (b) x3 - x + 1, 2.4 + x2 +x-5. -

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