Question

Can someone help in part D AND E PLEASE? solve it in general do not use numbers please

1. Let T: Pn(R) + Pn+1(R) be defined: T(P(x)) = (x + 1)p(x + 2) (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard bases {1, X, ..., x" } for Pn and {1, x, ..., xn+1} for Pn+1 if n - 4. (d) (5 marks) Let D : Pn+1(R) + Pn(R) be the derivative operator. What is the rank of DoT? Justify your answer. Describe ker(DoT). Is DoT one-to-one? (e) (5 marks) What is the rank of ToD? Justify your answer. Describe ker(ToD). Is To D one-to-one?

Answer 1

by d) soln Let T Pn (R) → Pnti (IR) be defined by TCP®)) = (x+1) ÞÓx+2) & the derivative operator :D :Pn+1 (R) → Pn (IR) be detined D (p(w)) – p'Coy the operator DOT: Pn (IR) Pn (R) Cube defined by DoT (PM) = D (ct12 BC+2)) 6C+1)Þ'x+27+b(42) the standard ordered basis, B- {1x, xh , &M of Pn (R) Then DOT (1) =(3+13.0 + [zi DOT(X) = (2+1). 1 +(x+2) DOT(X) = pc+i). 2(x+2) + (x+272 = 2x2+2x+4x+4 + x +4x+4 Considering 2x + 3 = 3% +10x +8

continuing 1 0 38 2 10 0 3, 0 - =1.2.3..ni m! +0.] p (2) EPnt they are lan find the Corresponding matrix (Dord as an upper triangular matrix of the from 6--1/+1)x(n+1) So, it's rank is ontl). Here, det (DOT) So, rank of DoT is nth). Chys. A at By Rank- Nullity theoren, ha dim (Pur(e))= R(DOT)+1(607)* (n+1) = (2+1) +N (DOT) ie ker (DOT)=6,0, 0, -- -0 (n+Otimes) Hence, DoT is one-one. ant (Aryf) (Aris's e) Again, the operator TOD! Phi (R) PnM (IR) bé destined by To D[p(x) = T(F"(*)) te. Too (p(x) = (x+1) Þ'(x+2), Now Cons the Standard ordered basis to = {1, x x xnantly of Pros (R) .: TOD(1) = 0, Top = (4+1), TOD (XY) = 64+1).2 (x+2) = 2x'+42+2x+4 > N(DOT) =0 => Ker (DOT) = 0 . ) (R)

2007 +6X+4 TOD (x²) (+1) 3(x+2) = 364+1)(x+4x+4) 3<3<3+4x74x + x +4474) - 3(x3 +5x478x+4) ie TOD (x²) 3343 +15x4+24X+]2. So, the the matrix of (TOD) & is of the form [ToD 131 0 1 © 1 0 O o 4 12 - 6 24 2 15- 0 3. 1 < 0 apper triangular full non-zeroq if is also a (+29X(n3) matrix of order (M+2) X (1+2). but here det (Toba) = 0x1x2x-xn=0, so, rank [ToD] b' (2+1) [ the last column of the matrix will be - rank of ToD is (nth) Any it we take the minor ..By Rank-Nullity theorem, excluding the list row Atst column, it will mon-zero dim ( Pnti (IR)) = Rank (TOD) + Nullity COD) 5) (7+2) = 6+1) + Nullity (TOD) Nullity (ToD) = 1 Thuy, kernel of ToD is a I dimensional Subspace of Pnti (IR) which satisfies (+U) Þ!(+2)for peng Kernel (TOD), (by'y And as by ③ Nullity (TOD)>0, Jo, Tod is not one-one, Any's be of