Question

Let T:P³ →P³ ​ be the linear transformation satisfying T(1)=4x² −9,T(x)=−3x−9,T(x² )=−5x² −x−6. Find the image of an arbitrary quadratic polynomial ax² +bx+c. T(ax² +bx+c)=

Answer 1

The image of an arbitrary quadratic polynomial ax² + bx + c under the linear transformation T is (4c − 5a)x² + (− a − 3b)x + (− 6a − 9b − 9c).

Step-by-step explanation:

To find the image of an arbitrary quadratic polynomial ax² + bx + c under the linear transformation T, we use the linearity property of T. That is, for any scalars a, b, and c, and any vectors v and w in the domain of T, the transformation satisfies T(av + bw) = aT(v) + bT(w). Given that T(1) = 4x² − 9, T(x) = −3x − 9, and T(x²) = −5x² − x − 6, we can determine the image of ax² + bx + c as follows:

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

T(ax² + bx + c) = a(−5x² − x − 6) + b(−3x − 9) + c(4x² − 9)

T(ax² + bx + c) = −5ax² − ax − 6a − 3bx − 9b + 4cx² − 9c

T(ax² + bx + c) = (4c − 5a)x² + (− a − 3b)x + (− 6a − 9b − 9c)

This is the resulting polynomial after T has been applied to ax² + bx + c.