Let Pk denote the vector space of all polynomials with degree less than or equal to k. Define a linear transformation T : P4! P3 by T (f (x)) = f (0) + f '1) (x-1) + f '' (2) (x-2) ^2+f''' (3) (x-3) ^3. Find the matrix representation for T relative to the standard basis {1; x; x^2; x^3; x^4} of R4 and the reversed standard basis {x^3; x^2; x; 1} of R3.

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Mathematics

20 August, 23:12

Let Pk denote the vector space of all polynomials with degree less than or equal to k. Define a linear transformation T : P4! P3 by T (f (x)) = f (0) + f '1) (x-1) + f '' (2) (x-2) ^2+f''' (3) (x-3) ^3. Find the matrix representation for T relative to the standard basis {1; x; x^2; x^3; x^4} of R4 and the reversed standard basis {x^3; x^2; x; 1} of R3.

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Francesca Shea · Answer 1

Step-by-step explanation:

T (1) = 1=0*x^3 0*x^2 0*x 1*1 T (x) = x-1=0*x^3 0*x^2 1*x (-1) * 1 T (x^2) = 2x^2-6x 6=0*x^3 2*x^2 (-6) * x 6 T (x^3) = 6x^3-48*x^2 141*x-141 T (x^4) = 24*x^3-204*x^2 628*x-604*1 collect the coefficient matrix and take its transpose

0 0 0 6 24

0 0 2 - 48 - 204

0 1 - 6 141 628

1 - 1 6 - 141 - 604