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)]B, where B={−1,−2x,4x2}

Please solve it in very detail, and make sure it is correct.

Answer 1

solution: given that T: P → P2 by Teatbr) = 3a - 26er + (a+b) n? 24 TC = 34 na T(n) = -2ntne the Matrin T o lo 2 - - R₂ → Rz - ₃ R . R₂ R3 + 1/4 R2 3. 0 -2 0 Basis of Range (T) le 23+0², -2ntn {[1]:67 (6) din (ker (T)) = 0 there baris of ker (T).

T is not onto because 3 - 2 & 4n2 E P but there does not enist at bn EP, such that T(a+bn) = 3cm +4422 because, cause, let Possible there enist such if R, (atbn) in then Tlatba) 13-n tyn2 3a - abnt (a + b) n2 a 3-nt 422 Comparty both sides we get a=1 b = 1 atb = 4 ire Iti ti = 4 - 3/2 = 4 which i contradiction

Id.) T in one one because Tea, & bp 2) = T(a + b2 ) for any (a, + bin) (a + b₂ ) EPI then. 2 3a, - abin & (9,7 b, ) n2 = 39 - 2b₂n + (a + b) By comparing both sides, sides, we get 19 =ą 2 and bi=bz a, + bin a tban do T is onl one [717+n)] [ 21 - 22 + 8 m2 B B 21- 2n +8n² =621).(-1) +1•(-2n) +2.1997 [T (7+2)]