Question

Let Poly3(x) = polynomials in x of degree at most 2. They form a 3- dimensional space. Express the operator Q(p) = xp' + p'' .

as a matrix (i) in basis {1, x, x^2 }, (ii) in basis {1, x, 1+x^2 } .

Here, where p(x) represents a polynomial, p’ is its derivative, and p’’ its second derivative.

Answer 1

Let Poly₃(x) = polynomials in x of degree at most 2. They form a 3- dimensional space. Express the operator Q(p) = xp' + p'' .

as a matrix

(i) in basis {1, x, x² },

(ii) in basis {1, x, 1+x² } .

Here, where p(x) represents a polynomial, p’ is its derivative, and p’’ its second derivative.

(i). We have Q(1) = x*0+0 = 0, Q(x) = z*1+0 = x and Q(x²) = x*2x+2 = 2x²+2.

Hence the matrix of Q relative to the basis {1, x, x² } is M = [Q(1),Q(x), Q(x²)] =

0	0	2
0	1	0
0	0	2

It may be observed that the entries in the columns of M are the scalar multiples of 1 and the coefficients of x, x² in Q(1),Q(x) and Q(x²) respectively.

(ii). We have Q(1+x²) = x*2x+2 = 2x²+2.

Hence the matrix of Q relative to the basis {1, x, 1+x² } is N = [Q(1),Q(x), Q(1+x²)] =

0	0	2
0	1	0
0	0	2