Determine which of the following sets is a subspace of Pn fo

Determine which of the following sets is a subspace of P_n for an appropriate value of n. All polynomials of the form p(t) = a + bt^2, where a and b are in R All polynomials of degree exactly 4, with real coefficients All polynomials of degree at most 4, with positive coefficients B only C only A and B A only

Solution

A :p(t)=a+bt^2

let p(t) =a1+b1t^2 and q(t)=a2+b2t^2 in A Nd c in R

Cp(t) + q(t) = ca1+cb1t^2+a2+b2t^2=(ca1+a2)+(cb1+b2)t^2

And this belongs to A as ca1+a2,cb1+b2 belongs R

A) is subspace of Pn

,B) polynomials of degree exactly 4

Not subspace of Pn as let x^4+1 and -x^4+x and c=1 in R then 1(x^4+1)+(-x^4+x)=x+1 that is not of degree 4

C) this is also not a subspace of Pn

Pollynomials of degree at most 4 with possitive cofe.

When multiplied by real scalar then will not necessarily be of possitive coeficient.