Question

3. Determine if each set is a subspace of the space of degree < 2 polynomials. If so, provide a basis for the set. (a) Degree s 2 polynomial functions whose degree 1 coefficient is zero: $(x) = ax2 + c where a,CER. (b) Degree s 2 polynomial functions whose degree 1 coefficient is 1: f(x) = ax2 + x + c where a,CER.

Answer 1

Subspace: ves a vector Space and so be a subset of rês called a Subspace of vis 2,465 +YES XES, CGR - CRES VRYES a -75 set Degree – 2, whose degree 1 co-efficient is rew. Par ante, a, CER f(2), 8x) E To the go set P(x)= a ² + c can, CGR gex) = A x2 + C2 199 + ) Ka, + 9) +(+1) Prence Sca) + gen) belongs to the set PER P. f(x) = (pai) x + (pe belongs to the Hence this forms a Subspace. Ds. Degree < 2, 51- axat c} 2 (a), a) ES =(x)= a, x² + x + C, where a) = A₂2² + x + c auf f(x) + 9a)= a1 +92)2 + 2x + Since co-efficient of degrees is 2 et @ GR @ 71 c. f(a)=cadx²+ (GX+(C. C) Then coefficient of degree 1 à 4 & 1 C. SAYS Subspace a, c, 92, C, GIR 2 (6+ (2) 75 Nota