Do the polynomials p 1 Do the polynomials p1 t 2 p2t 1 t for

Do the polynomials p

1) Do the polynomials p1 (t) 2, p2(t) 1 t, form a basis of Ps? Why or why not? (2 points)

Solution

We need to check if they can generate the standard basis vectors of P5 ie 1,t,t^2,t^3,t^4,t^5

1=-p1/2 , so 1 can be generated

t=p2-1=p2+p1/2, so t can be generated

t^2=p3-t=p3-p2-p1/2, so t^2 can be generated

t^3=p4-t^2=p4-p3+p2+p1/2, so t^3 can be generated

t^5=-p5+t^3, so t^5 can be generated

But t^4 cannot be generated

Hence it is not a basis for P5