The function F, given by the formula F(a + bx) = 7(a - b) -5(a - b)x is a linear operator from P1 to P1. Find and simplify the following: F(10 + 5x) 35 - 25x F(11 + 3x) 56 - 40x Part 2 of 5 Write the matrix MB( F) for this operator, with respect to ordered basis B = {1, x}. Answer: Hint MB(F) = 7 -7 -5 5 Part 3 of 5The number A] = U is an eigenvalue for this operator. Give three different non-zero polynomials that are eigehvectors corresponding to this eigenvalue. Give your answers in simplified form. Hint Compare coefficients 1 and n. we get, Fa- 76 = 120 -(1) and - 59+ 56= 126-(1) () 79-120 - 76= 0 7-59- 76=0 1 |a = - 7 b 5 and cij J -50+ 56- 12b= 0 7-50 - 7b=0 + Taz - 7b 5 .. Eigenvectors will be of the form v=- 76+bx 5 ( " a= - 7 6) Take b = 5, then 5 P(20)= 18 = -7+5x1. Take b= 10, then Plac1=V = -14+lox Take b = 15 , then Poll=0 =- 21 +15xIf 1=la, then by the definition of eigenvalue and eigenvector: F ( 1 1 = 10 Tofind: V= atbx 1 Flatball= 12 ( a+ by) vie. to find value of a and b 17 ( a-b)- 5 (a-b)a= 120+126x Here, Vto. 7 7a-76-507+ 5bXX= 12a+labx . : V is an eigenvector 7 0 - 7b) + (- 59+5b)X = laat labx