Let T P2 implies P2 denote the following linear transformati

Let T: P_2 implies P_2 denote the following linear transformation T (ax^2 + bx + c) = a(x - 1)^2 + b(x - 1) + c. Let alpha = {1, x, x^2} and beta = {x^2 - 1, x^2 + 1, x + 1}. Find [T]^alpha_alpha, [T]^beta_beta and [T]^beta_alpha.

Solution

Solution: Given that T: P2 \\to P2 is the linear transformation defined by

T(ax² + bx + c) = a(x-1)² + b(x-1) + c = a(x²-2x+1) + bx - b +c

= (a - b + c) + (b - 2a)x + ax²

Here \\alpha = { 1, x, x²}

Now T(1) = 1 + 0.x + 0.x² = 1 (as a=b=0, c=1)

T(x) = -1 + 1.x + 0.x² = -1+x (as a=c=0, b=1)

T(x²) = 1 + (-2).x + 1.x² =1-2x + x² (as a=1, b=c=0)

So matrix representation of T with respect to basis { 1, x, x²} is

[T]_{\\alpha}^{\\alpha}=

Here \\beta ={x²-1, x²+1, x+1} is another basis of P2.

Now T(x²-1) = (-1-0+1) +(0-2)x +x² = x²-2x = -(1/2)(x²-1) + (3/2)(x²+1) - 2(x+1)

T(x²+1) = (1-0+1) +(0-2)x +x² = x²-2x+2 = -(3/2)(x²-1) + (5/2)(x²+1) - 2(x+1)

T(x+1) = (1-1+0) +(1-0)x + 0.x² = x = (1/2)(x²-1) - (1/2)(x²+1) +1(x+1)

So matrix representation of T with respect to basis {x²-1, x²+1, x+1} is

[T]_{\\beta}^{\\beta}

Again

T(1) = 1 + 0.x + 0.x² = 1   = (-1/2)(x²-1) + (1/2)(x²+1) + 0.(x+1)

T(x) = -1 +1.x + 0.x² = -1+x = (x²-1) - (x²+1)   + (x+1)

T(x²) = 1 +(-2).x + 1.x² =1-2x + x² = -1(x²-1) + 2(x² + 1) -2(x+1)

So matrix representation of T with respect to basis { 1, x, x²} and {x²-1, x²+1, x+1} is

[T]_{\\alpha}^{\\beta}

1	-1	1
0	1	-2
0	0	1