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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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Problem Set 1 Ql which of the following are innerproducts 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 innerprod uct on W. Show that the function <,>' defined by < rc,! )' :< T(x),Kil > is an innerproduct iff Zis one-to-one. Q3 a)Use the Cauchy Schwaz inequality in C' (with the standard innerproduct ) 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: f n f * l n - t + f n r * l E ' a ; * b ; t z | = l l t a , 1 2 | . l l r b i r 2 I L,=r I L,:r I Ler J Q4 Which of the following matrices are orthogonal Q5 Let (V,<,>) be an lPS. a) Letv e Vand supposethat( x,v >:OVx e V. Showthatv - 0. b) Letxoy € Vand suppose that< x,y >-< y,v ) Vv e V. Showthatx : y. 1 0 0 0 o + ; 0 o + o r 0 + + 0 ':[: Ii ],:[ :+*]

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Problem Set 2 Q1. Gonvert the following bases (for the respective subspaces)
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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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