Question

Let T:P2​(R)→P2​(R) be defined by T(a+bx+cx2)=(3a−b+ c)+(a−c)x+(4b+c)x2, and let β be the standard ordered basis in P2​(R).

(a) Prove that T is an isomorphism.
(b) Find [T]β​, the matrix representation of T in the basis β. Prove that T is an Isomorphism.
(c) Find [T−1]β​ Using theorem 2.18 (You need to find the inverse of the matrix [T]β​ using any known to you method.)
(d) Find T−1(a+bx+cx2) using matrix [T−1]β​ and Th. 2.14
(e) Find T−1(2−x+2x2).

Answer 1

The question involves proving that a linear transformation is an isomorphism, finding the matrix representation of the transformation and its inverse, and using the inverse to find the image of specific polynomials within P2(R).

Step-by-step explanation:

Proof and Calculations for Isomorphism and Matrix Representations

The question relates to linear algebra, specifically regarding linear transformations and their matrix representations. A student is asked to demonstrate that a given transformation T is an isomorphism, to find the matrix representation of T with respect to the standard basis in P2(R), calculate the inverse of this matrix, and finally, to apply the inverse transformation to a specific polynomial. The transformation T maps a polynomial a + bx + cx^2 to another polynomial (3a-b+c) + (a-c)x + (4b+c)x^2 within the same space. To prove that T is an isomorphism, one must show that it is bijective, i.e., both injective (one-to-one) and surjective (onto).

The matrix representation of T with respect to the standard ordered basis β = {1, x, x^2} is determined by applying T to each basis vector and expressing the result as a linear combination of the basis vectors. This yields [T]β, a 3x3 matrix that represents T in the standard basis. To find the inverse matrix [T^-1]β, one can employ various methods such as Gaussian elimination or finding the adjugate and determinant to calculate the inverse. Finally, to find the image of a specific polynomial under T^-1, one can use the inverse matrix and matrix multiplication to compute the coefficients of the resulting polynomial.