Question

2. (a) Let B = {f1, f2, f3} be a subset of P2 where fi(x) = x² – 3, f2(x) = x2 – 2x and f3(x) = x. Show that B is a basis of P2. (b) Determine whether or not the following sets are subspaces of F. (i) X = {f € F | f(x) = a(x + cos x), a € R}. (ii) Y = {f EF | f(x) = ax + sin x, a € R}. (c) Let {u, v, w} be a set of linear independent vectors in a vector space V. Determine whether or not the set {u, u +5v, u - v - w} is also independent in V.

Answer 1

Solution page a @ f(x) = x²3 f (a)= 2² zx, f(x) = x + claimi B = {fi, fa, f₂} forms Basis for P₂ More dim (Pn na dim ch 1=3 Now we have to show that only espan {B} P2 Becaure. If vis veetor space of dimension n 4 B sv with Bl=n4 span {B}=v then B forms Basis for v. Now, fca) + 2 f (0) = 2² f(x)=x 3 + [ f, CV) - (fx ca] +2fz (x))] = 1 -) {1, 2, x23 c spa n {B} a span { 1, 2, 223 ç span {B} span {B} = P2 Becaure {\, &.X2Z is standard basis of P₂ A B forms Basis for P2

6 Page- i) X= {fff I fox) = a (atcoux), aer} Step-II o= ocatoru) 4 OER = fcasso (zero function) is inx =) X 90 Step-I let fi fa EX = f, cal = a, cateosa) 4 fex) az cateosa) for some a,, an AR a. f, co - 2₂ CM) = Cara) catcom) het aa a, az a, a ER =) a, d = a ER f. cx) - f₂cx) G as well as in X le fick) - hen ex x is subspace of F.

Rag ii) Y = {f efl fon) = axtsina, at R} But faer sot. aatsina tea) & Y (remo function) => Y is not subspace of f. {", W, W} be linearly independent let veeters in a veeter space V. over F claim ar cheeking { ", utsy, unv-w} is lin linearly independant on not. س ملر consider *, ,&₂, kg Ef soto Ricult de cat5V) dz 64-V-W) = 0 Lu tha utslaut az u -LzV - 2 W=0 cait de talut (52₂-23) + (-2₂) W=0 But . {uv, w } ir linearly independant set Litch th=0 54-23=0 - a320 =) L₂=0 = &z=0 562 -0 =0 -) 2,=0 & toto=0 satu, U-v.W } is Linea zly independant set.