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**preview**shows page 1. to view the full**4 pages of the document.**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, t2−1}, 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,c∈Rand 1 ≤j≤3.

b) Let f:V→Rbe deﬁned by f(p(t)) = p(2) + p0(0), p(t)∈V. (Here, p0(t) is the derivative

of p(t).) Show that f∈V∗. 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 V∗be the dual space of V. Deﬁne

f1,f2and f3∈V∗by:

f1(a+b t +c t2) = −a+b, f2(a+b t +c t2) = 2 a+b, f3(a+b t +c t2) = −b−c, 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 V∗is 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 V∗that is dual to β. For 1 ≤j≤nlet yj∈Vbe such

that fj(x) = hx, yji,x∈V. 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:V→Vbe linear. If x∈V,

deﬁne fx∈V∗by fx(y) = hy, xi,x∈V. Prove that the adjoint T∗of Tis the unique linear

operator on Vthat satisﬁes Tt(fx) = fT∗(x)for all x∈V.

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 y∈H

such that xy−1/∈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={T∈GL(V)|N(T−IV)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={A∈G|A2=I}.

d) Let G={A∈GL3(F)|A21 =A31 =A32 = 0 }(that is, Gis the group of upper

triangular matrices in GL3(F)). Let H={A∈G|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.

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