Find a basis B for the span of the given vectors. B = [01-41] [51-10] [4191] 11

5

Consider the following polynomials as vectors from the vector space P3 (R) (Definition 2.13 in the Sheldon's book): p(z) = 2³-2z+1 q(z) = 2²-1 r(z) = 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 + (az-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)? What is span(q, r, s)?

4. Find a basis for the span of {[1, 2, 1]", [3, 1, –1]", [1, –3, –3]"}