Question

If p(t) = a + t² is the form of all the polynomials in S, where a is some real number. Is S a subspace of P sub {2}.

Answer 1

Doesnt this depend on the value of a, if a<= 0 then the zero vector will be in S but if a > 0 then 0 vector wont be in S and it wont be a sub space?

Answer 2

Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following three conditions:
The zero vector, 0, is in W.
If u and v are elements of W, then any linear combination of u and v is an element of W;
If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W
Take a polynomial in S:
\[P_3(t)=3+t^2; P_5(t)=5+t^2\]
For it to be a subspace then it must contain the zero vector. It does if a=0 since t^2 goes through the origin (I believe).
However, any LINEAR COMINBATION of P_3 and P_5 and P_N should give an ELEMENT OF S (since thats the "subspace"ity we're checking) Notice though that: \[P_3(t)+P_5(t)=8+2t^2\]
THE COEFFICIENT ON THE t^2 IS NOT 1 AS REQUIRED BY THE DEFINITION OF THE SPACE. Not a subspace. (If I'm understanding what you mean by p sub 2.