Question

U = {pe P.(R) p(e)dr = o} (a) Find a basis for U. Justify that your answer is a spanning set and is linearly independent. MAT310 SPRING 2020 - HOMEWORK 2 (b) What is the dimension of U? (€) Find a subspace W of PA(R) such that P4(R) = U W.

Answer 1

And wen Uz/per, () : "Spe) onzo? To find the basis of : We know in general, that Pu(x) was the basis {1,2, 22, 23, 24} which is the Handard basis of Pu(x). do im Py (o) 25 and u contains p(x) 6 Pu (ad sit- /p(6) 20 roles. Now, let us anome that f(n)= auta, 27 422 & 923 EP, (R) be an arbitrary polynomial. let p(x) E U. 1. tagnes )zo holds. ->/Cart 9,34+, may saya") 02 20 bay 2) Taonfa, nt aut 32457/ 5 / 20 z) 24. + 2q + 2 any zo 2) art antal zo 2) ao 2-( auto -. : p(x) can be written as - - ( ) + 7,4*444'7 azert 2,39

..qx, 22, 23, n7 is a basis of U, which is the subset of L.I. set 26, 2, 2, 53,148 nence fox, n², x3, ny} is L.I i.e. linearly Independent. and obviongly it is the genearating set of Usince any arbitrary dement p (n) of o can be written as the linear combination af 20, 22, 23, nh] with the preop. 6. 2. {x, 2², 23, de } becomes a basis of O. ② Since Per (12) nas dimension 5. and 't has a constraint with (P (n) = 0, hence dim 0 5 -1) 24.1 Also in @ we proved this since 12, 22, 93, at} is a baris of U. © If we consider a subspace Wzis that hoitw contains all the scalars, then Pu (2) can be written as - Pu (UX) = vOW. Hence the proof