Complete the set S = {x³+x²+1,x²+x+1,x+1} to get a basis of P3 ?

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Answer 1 · 2018-01-30T17:54:04

See below.

Considering the canonical basis

#{1,x,x^2,x^3}# we have

#((1,0,1,1),(1,1,1,0),(1,1,0,0),(alpha,beta,gamma,delta))((1),(x),(x^2),(x^3))#

generates any #p_3(x)# polynomial iif

#det((1,0,1,1),(1,1,1,0),(1,1,0,0),(alpha,beta,gamma,delta))=alpha-beta-delta ne 0#

thus for instance, for #alpha=1,beta=1,gamma=0,delta=1# we have that

#{(p_0=x^3+x^2+1),(p_1=x^2+x+1),(p_3=x+1),(p_4 =x^3+x+1):}#

is a generator basis for all #p_3(x)#