ass7 - P roblem et 1 S Ql w hich o f t he f ollowing re i...

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Problem Set 1 Ql which of the following are inner products on the given spaces? a) T : R2 < (x,y),(u,v) >: 2xu + w +W + Zyv b) V : Rz < (x,y),(u,r) >= xu + ?:cv + zyu + W c) It : Mz"z(R) < A,B >= Trace(A + B) dl It: Mz"z(R) < A,B >: Troce(B*A) e\ V : Pz(R) I pr,pz >= p{A}.p2(t) +p1e).pz(O) f) Y : Pz(ft) < pt,pz >= p{0).p2(0) +pr (1).p2(1) + p{Z).pz(Z) Q2 Let T : V -> W be a linear transformation and let <, > be an inner prod uct on W. Show that the function <,>' defined by < rc,! )' :< T(x),Kil > is an inner product iff Zis one-to-one. Q3 a)Use the Cauchy Schwaz inequality in C' (with the standard inner product ) to prove that: ,fo,6, = [f ,0; t2l=[i , a, ,, I Fl L r=l J L;=r J b) Use the triangle inequality in C' (with the standard inner product ) to prove that: fn f* ln -t+ fn r* lE'a;*b;tz | =llta,12 | .llrbir2 I L,=r I L,:r I Ler
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This note was uploaded on 10/21/2010 for the course MATHEMATIC stats taught by Professor Various during the Spring '10 term at Waterloo.

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ass7 - P roblem et 1 S Ql w hich o f t he f ollowing re i...

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