Transcribed Text
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 ntuple 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) (xAn).
(b) Prove that the minimual polynomial of T is II(xAi), 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 +x5. 
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