Question

Let T:P1→P2T:P1→P2 be a linear transformation defined by

T(a+bx)=3a−2bx+(a+b)x2.T(a+bx)=3a−2bx+(a+b)x2.

(a) Find range(T)range(T) and give a basis for range(T)range(T).

(b) Find ker(T)ker(T) and give a basis for ker(T)ker(T).

(c) By justifying your answer determine whether TT is onto.

(d) By justifying your answer determine whether TT is one-to-one.

(e) Find [T(7+x)]B[T(7+x)]B, where B={−1,−2x,4x2}B={−1,−2x,4x2}.

Answer 1

Range (T) = { T (atba): atba EP, : atba 3 a-aba + (atba" ce ? (3+2) a (3+2) a + (-2x+2²) ashari = }(348) a + (-22+x+) b: a belry Range (t) = span span{ (3+x), (-2x+x)} . dem Range (t) = 2 (** (3+2?), (+23+7) are linearly Basis of Range (T) independent) { sta", -2*****

0 kort (at&x) EP, T (arba) -o? { (arba) EP, : (8 + x) a + (-2x + x) (atba) EP, : b(2x-2² 3+22 But then , at bu b(22-0) br + 3+02 27.-22 + ( 3+x2) ? 3 taz i for a to, bto which is not a polynomial in P, if a=bzo, Then there to oth a=o, b=0. Ker T = 6208 atbx=0 : dim ker t = 0. Basis of kort=p. dim P₂ = 3 But din Range (T) = 2 " . T is not onto dim ort=0 kur (T) = 204 T is One-One

21-qat 8x2 [1 (10) B = -1, -2*, 4* 1440) T (74x) = 3(7) -2(1) x + (7+1)x² B - (_)(21) + (-2x) 42 (470) [+(7+*)] O").