Consider the following polynomials as vectors from the vector space P3 (R) (Definition 2.13 in the Sheldon's book): p(z) = 2³-22+1 q(z) 2²-1 = r(2) 22³-2 = 2²-22+1 = t(z) = 2³ +az²+b, z € R. (When asked about span of a list of vectors, you are expected to express it in the following form: span(p, q) = {a32³ + a₂2²-2a3z + (a3-a2) € P3(R) : a3, a2 € R}.) (a) Are the vectors p, q, r linearly independent? (b) Are the vectors p, q, r, s linearly independent? (c) * For what values of a € R and b ER the vectors p, t linearly independent? (d) For what values of a R and b E R the vectors p, q, t linearly independent? (e) For what values of a € R and b ER the vectors p, q, r, s, t linearly independent? (f) What is span(q, r)? (g) What is span(q, r, s)?

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