4) Let $P_3 = \{a_0 + a_1t + a_2t^2 : a_0, a_1, a_2 \in \mathbb{R}\}$, i.e., all polynomials of degree less than three. Note that $P_3$ is a vector space. Let $p(t) = t + 1$, $q(t) = 2t^2$, $r(t) = t^2$, which are all elements of $P_3$.
(a) Determine if $p$ and $q$ are linearly dependent or independent.
(b) Determine if $p$ and $r$ are linearly dependent or independent.
(c) Determine if $q$ and $r$ are linearly dependent or independent.
(d) Do $p$ and $q$ form a basis of $P_3$? (If yes, just write as much. If not, explain why not and provide a set of elements of $P$ that do form a basis.)
(e) What is the dimension of $P_3$?
