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Lecture 1

MAT247H1 Lecture Notes - Lecture 1: Dihedral Group, Linear Map, Group Homomorphism

Department
Mathematics
Course Code
MAT247H1
Professor
Fiona T Rahman
Lecture
1

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MAT 247S - Problem Sets 8–9
Problem Set 8(questions 1–15), due Thursday April 2; questions 3b), 4, 8b), 10a) and 15b)
will be marked.
Problem Set 9, questions 16–29; not to be handed in.
1. Let Tbe a linear operator on a ﬁnite-dimensional complex vector space. Suppose that Tr+2 =
4Trfor some positive integer r. Prove that Tis diagonalizable if and only if T3= 4 T. (Hint:
What can you say about the minimal polynomial of T?)
2. Let V=P2(R). Let β={t+ 1, t 1, t21}, and let β={f1, f2, f3}be the dual basis for
V.
a) Compute the values fj(a+b t +c t2) for a,b,cRand 1 j3.
b) Let f:VRbe deﬁned by f(p(t)) = p(2) + p0(0), p(t)V. (Here, p0(t) is the derivative
of p(t).) Show that fV. Express fas a linear combination of the basis vectors f1,f2
and f3.
3. Let V=P2(R) = {p(t) = a+b t +c t2|a, b, c R}. Let Vbe the dual space of V. Deﬁne
f1,f2and f3Vby:
f1(a+b t +c t2) = a+b, f2(a+b t +c t2) = 2 a+b, f3(a+b t +c t2) = bc, a, b, c R.
a) Prove that {f1, f2, f3}is a basis for V.
b) Find the basis β={p1(t), p2(t), p3(t)}for Vhaving the property that the dual basis β
for Vis equal to {f1, f2, f3}.
4. Let Vbe a ﬁnite-dimensional inner product space. Let β={x1, . . . , xn}be a basis for V, and
let β={f1, . . . , fn}be the basis for Vthat is dual to β. For 1 jnlet yjVbe such
that fj(x) = hx, yji,xV. Set β0={y1, . . . , yn}.
a) Prove that β0is a basis for V.
b) Prove that β0=βif and only if βis orthonormal.
5. Let Vbe a ﬁnite-dimensional inner product space, and let T:VVbe linear. If xV,
deﬁne fxVby fx(y) = hy, xi,xV. Prove that the adjoint Tof Tis the unique linear
operator on Vthat satisﬁes Tt(fx) = fT(x)for all xV.
6. #5, §2.6
7. # 7, §2.6
8. Determine whether His a subgroup of G. If His not a subgroup of G, ﬁnd xand yH
such that xy1/H. If His a subgroup, answer the following two questions (and justify your
answers): (i) Is Han abelian group? (ii) Is Ha normal subgroup of G?
a) Let Vbe a ﬁnite-dimensional vector space and let G=GL(V) be the group of invertible
linear operators on V(with the group multiplication given by composition of operators).
Let H={TGL(V)|N(TIV)6= 0 }.
b) Let G=GL2(Q) and H= a3b
b a |a, b Q,at least one of aand bis nonzero .
(Here, Qis the ﬁeld of rational numbers.)
c) Let G=GL2(F) and H={AG|A2=I}.
d) Let G={AGL3(F)|A21 =A31 =A32 = 0 }(that is, Gis the group of upper
triangular matrices in GL3(F)). Let H={AG|A11 =A22 =A33 = 1 }.
e) Let Gbe the group of upper triangular matrices in GL3(F) and let Hbe the set of all
diagonal matrices in G.