(Solved): (5 points) Consider the vectors p1(t) = 1 + t2, p2(t) = t + t2, p3(t) = 1 + 2t + t2 in the vector sp ...

(5 points) Consider the vectors
p1(t) = 1 + t2, p2(t) = t + t2, p3(t) = 1 + 2t + t2
in the vector space P2 of all real polynomials with degree at most 2.
Use the theory of coordinate vectors (or any other method) to answer the following questions:
(a) Do the polynomials p1(t), p2(t), p3(t) form a linearly independent set in P2?
(b) Can the polynomial p(t) = 1 + 4t + 7t2 be written as a linear combination of the polynomials p1(t), p2(t) and p3(t)?

(5 points) Consider the vectors \[ \mathbf{p}_{1}(t)=1+t^{2}, \quad \mathbf{p}_{2}(t)=t+t^{2}, \quad \mathbf{p}_{3}(t)=1+2 t+t^{2} \] in the vector space \( \mathbb{P}_{2} \) of all real polynomials with degree at most \( 2 . \) Use the theory of coordinate vectors (or any other method) to answer the following questions: (a) Do the polynomials \( \mathbf{p}_{1}(t), \mathbf{p}_{2}(t), \mathbf{p}_{3}(t) \) form a linearly independent set in \( \mathbb{P}_{2} \) ? (b) Can the polynomial \( \mathbf{p}(t)=1+4 t+7 t^{2} \) be written as a linear combination of the polynomials \( \mathbf{p}_{1}(t) \), \( \mathbf{p}_{2}(t) \) and \( \mathbf{p}_{3}(t) \) ?