let us consider following problem
A subsepace $S$ of a vector space $V$ is given.
Determine a basis for $S$ and extend your basis for $S$ to obtain a basis for $V$.
$V=P_2$, $S$ is the subspace consisting of all polynomials of the form $$(2a_1+a_2)x^2+(a_1+a_2)x+3(a_1-a_2).$$
we know that for polynomials of order $2$ basis can be represented by following set of vectors
$(1,x,x^2)$ now we want to extend this basis for vector space $V$,does it means that for vector space $V$ linear combination of basis of $S$ will be another basis or?clearly if $V$ is vector space of order more then $2$,we can add more element like $x^3$,but in this case what we should do?thanks in advance