Consider the linear transformation from the space of degree

Consider the linear transformation from the space of degree or two less polynomials to itself defined by: D (ax^2 + bx + c) - 2 ax + b + (a/2)cx^2 (You may take it as given that this actually represents a linear transformation). Write down the representation of this transformation with respect to the basis (1, x, x^2). Then write down the representation of the transformation with respect to the basis (6, 6x, 3x^2).

Solution

Let T denote the given linear transformation.Then T(ax²+bx +c) = D(ax²+bx +c) -2ax +b +(c/2)x².

Hence T(1)=D(1)+x²/2=D+x²/2 (here, a=0=b and c =1). Also,T(x)=D(x)+1= Dx +1(here, a=0=c and b =1), and T(x²) = D(x²)-2x = Dx²-2x(here, a =1,and b=0=c). Since the matrix representation (A) of a transformation has columns which are images of the ve4ctors in the standard basis of the domain, hence A =

D

1

0

0

D

-2

1/2

0

D

Similarly, T(6) = 6T(1)= 6(D+x²/2)=6D+3x², T(6x) = 6T(x) = 6(Dx +1) = 6Dx +6 and T(3x²) =3T(x²) = 3(Dx²-2x) = 3Dx² -6x . Hence the matrix representation of T with respect to the basis (6,6x,3x²) is B =

6D

6

0

0

6D

-6

3

0

3D

D	1	0
0	D	-2
1/2	0	D