Denote by PnR the space of all polynomials with real coeffic

Denote by P^n(R) the space of all polynomials with real coefficients up to degree n. Consider the linear map T : P^2(R) Rightarrow P^2(R) given by Tp(x) = x^2 d^2p/dx^2. Find the matrix representation [T]_alpha^alpha of T with respect to the basis alpha = {x, 1 - x, 2 + x^2}.

Solution

Here n = 2 ,so

let p (x) = ax² + bx +c

T p(x) = x² *(d²p/ dx² ) = x² * 2a (dp/dx = 2ax + b , hence d²p/ dx² = 2a)

T (x) = 0 (here a =0, so x² * 2a = 0)

T (1-x) = 0 (here a = 0 x² * 2a   = 0 )

T (2+x²) = 2*x² (here a = 1 x² * 2a   = 2*x²   )

Now we have to represent 0, 0 , 2a * x² in terms of basis {x,1-x,2+x² }

lets do it for any general quadratic polynomial (ax² + bx +c )

let c1,c2,c3 be coefficients of corresponding basis

so the equation is   

ax² + bx +c = c1 *x + c2 *(1-x) + c3* (2+x²)

solving we get

c2 + 2*c3 = c (equatting the constant )

- c2 = b (equating the coefficient of x)

c3 = a   (equating the coefficient of x²)

so c1 = -2a+b+c, c2 = -b, c3 = a

now for 0 ,a = b = c= 0 , so c1 =c2 = c3 = 0

for 2x² a = 1, b = c = 0, c1 = -4 , c2 = 0 , c3 = 2

so the representation of matrix T =

0	0	-4
0	0	0
0	0	2