Let T P3 rightarrow P3 be the linear transformation satisfyi

Let T: P_3 rightarrow P_3 be the linear transformation satisfying T(1) = 2x^2 -4, T(x) = 2x + 9, T(x^2) = -5x^2 + x - 7. Find the image of an arbitrary quadratic polynomial ax2^ + bx + c. T(ax^2 +bx + c) =.

Solution

T(ax² + bx + c) = aT(x²) + bT(x) + cT(1)

=a(-5x^2 +x-7) +b(2x+9) +c(2x^2 -4)

=(-5a+2c)x^2 +x(a+2b)+(-7a +9b -4c)