Question

2. Consider the inner product space V = P,(R) with (5.91 = 5(0)g(t) dt, and let T: VV be the linear operator defined by T(f) = xf'(x) +2f(x). (1) Compute T*(1+2+x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.

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Answer 1

Answer :

Given data :

Inner product space $\bf V= P_2(\mathbb{R})$ $\bf \langle f,g\rangle=\int_{-1}^1f(t)g(t)dt$

and

$\bf T(f) = xf'(x) +2f(x)$

So, we will find the values of T(1), T(x) and T(x²) from given data to find the answer of part (i).

Similarly we have to find the value of $\bf \beta$ in part (ii).

I have made some handwritten notes for step by step answer.

Please go through them.

Page Date: Notes Answer: Giron data :- 1 Innes product space V = Po (R) <f. g) = I fit) get) dt Tlf) = nfin tafo) - a > linear operator, (aSo, from equation ② For basis of V= {1,24 223 for food = 1 f'(a)= 0 :T(1) 3(0) f ( 1 ) T (1) = 2 =) (9x1)+ (Oxa) + (Ox) - Similarly for flo=n 3 f(2)= Tlou) = R(1) + 2 101) T(n) =30 → (011) + (3x0) + (0x029) şimilarly for from = 2² flon) = 20 T( a) = a - () 3 ( 3 ) TR²) = 42² >> (Ox1)+ (oxo) +(4xm') -

Notes Pate Duit So, for {1, M, "} bass of V, from ea? 3, 4 and 5. I matrix 2 Oo 3 o 0 4 co I has all real elements and it is diagonal madox So from eqn 6 Т. T $0 ind T T T 9 6 2 O 0 3 o 3 0 4 o 0 4 = T* - T It means (rs) to cause T ( 1+n+ m2) T(1+0+2²) Tescolt (GE) (10) E I to is given that T is a linear operator So Th (1+ 2+2) = T (1+ 2+72) T (1) + T(R) P + T (o Toacher's Signature

Notes Page: Data: putting valuess from from ea eqr ③ 1 and ② in eq" @ Th ( 14 N + N²) 2 + 3 + 42² 0 ortho normal we To find. if there are baris of eigen vectors ß. So, by I is diagonal and I has can See that n q" @ We can see that, eigen valves 2, 3 and 4. from ea" ② 9 and that a . we can see and Trai= Yxge 1 respectively T(1) = 2x1 T(2) = 3xn has Eigen rectors, ชา and So In this case Orthonormal baru - C. N L 11111 11n11 ß-S 1 7 n? 110'll 10

Notes We know that 1 <,> where <f. f> : f.f da 1 0 so ( TELE S fif dar ? The 1/2 1 -1 Then from eq" 0 11111 = 1. 1) da 2 = 1 1 12 S fdn -1 1 162 sal S2 2 Elv 2 -1 from equ 1 101) 2: 1.2) da 39: Salda? 1 1 112 I nr 4 2 3 $33 13

Page: Data: Notes n from ea nr?)) = f(no no I da 11 14) { 25 5 10 117 { 2 Putting valves of equ W 13) and (14) ( In eq" (1 we get B = C N (VB) 2 3 So Orthonormal bases : f = { 1 / 1 pa 15 N² 2 1 N ادارہ 2 2 2

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