Question

Show that each of the following subsets are not subspaces by finding a counterexample. (a) The set of polynomials of degree exactly 2, as a subset of P. (b) The set of polynomials p(q) in P, such that p(1) = 1, as a subset of P. (c) The set of sequences with non-negative terms, as a subset of S.

Answer 1

© Set of polynomial of degrees exactly 2 s={ poil deg (P) = 2 } set degcp) = 2 deg(P)= 2 deg (P2)=2 deg Pi+P2) = dog (P. 3 + deg (P2) = 2+2 = 4 #2 so s is not seebspace + soin let b) The set of polynomial place in P2 =S such P(i)=1 PIC1) = 1 P2 (1)=1 (Pi+B)( 1) = PIUS + P2612 ( Linearity is not hold) = 1 + 1 = 2 = 1 so S es not selbspace €) sayset of sequences with non negative number term s={ 1,+1,1,1,1,1 - ...} t a term > 0 d=-1 ER 28 = (-1,-1,-1,4,-1,-1---) : XeR >0 > tve so s is not subspace