Question

Let T: P1 → P2 be a linear transformation defined by T(a + bx) = 3a – 2bx + (a + b)x². (a) Find range(T) and give a basis for range(T). (b) Find ker(T) and give a basis for ker(T). (c) By justifying your answer determine whether T is onto. (d) By justifying your answer determine whether T is one-to-one. (e) Find [T(7 + x)]], where B = {-1, -2x, 4x2}.

Answer 1

Giren 1 (ad on Tlat bu) : 3a - 2bu Put ad bn . y y- - T (4) 3a - 28ly-a) 30 Cant) (93) ray) My) = ( 4 ) 3 23(0,333 065) ryt zad Qb :) (y² + a? 6 t a 55+ in the form of py2 + gyar Tly) is So if > p = atb > Range (*) ) If P ad 6 62 Т. 4*)] » Range (-00 (b) TO find ken (T) - py 2 od ayar Ziyo we can And After that, accordingly find ken(7). determined the range of No T is indo. As in section (a), we Tly), and it is not 'R' So T is not onto

(d) is not one to one AS T a is quadratic polynomial. will be 2 Values (equal) of Tly) cat two values of y. having le) T(7+h) bi :. 1940 2x11 t ( 713 72 24 21 842