Show that tˆ2 + t + 1, tˆ2 + t + 2 are linearly independent in P2. Which of the vectors 1, t, tˆ2 can be added to those two polynomials in order to obtain a basis for P2 ? If there is more than one of these three vectors that works, then provide all of them.
To show that the two polynomials are linearly independent in P2, there must be scalars let's say a1 and a2 of the field F such that the linear combination: a1*(t2 + t + 1) + a2*(t2 + t + 2) = 0 iff it suffices to take a1 = a2 = 0 Regarding the task to obtain a basis for P2, the set of vectors must span the vector space P2 and must be linearly independent. I tried adding each vector to the two polynomials (such as a1*(tˆ2 + t + 1 + tˆ2)+ a2*(tˆ2 + t + 2 +2), however, it does not work to prove they form a basis. Thanks for your assistance.
Partial Solution / Hint: Let's call $p_1 = t^2 + t + 1$ and $p_2 = t^2 + t + 2$. Then you notice that $p_2 - p_1 = 1$, so $1 \in \operatorname{span}\{p_1, p_2\}$, so you would gain nothing in the span by including $1$ (since it is already in the span). Now, since $1$ is in the span, then $p_1 - 1 = t^2 + t$ is also in the span of $\{p_1, p_2\}$, so if we were to include $t$, say, then $(p_1 - 1) - t = t^2$, which means that $\{t, p_1, p_2\}$ have $1, t$, and $t^2$ in their span.