(a) Prove that the polynomials

i = 1.........k, are linearly independent if and only if the k × (n + 1) matrix A whose entries are their coefficients aij ,1 ‰¤ i ‰¤ k, 0 ‰¤ j ‰¤ n. has rank k.
(b) Formulate a similar matrix condition for testing whether another polynomial q(x) lies in their span.
(c) Use (a) to determine whether p1 (x) = a3 - 1. p2(x) = x3 - 2x + 4. p3(x) - x4 - 4x, p4(x) = x2 + 1, p5(x) = - x4 + 4x3 + 2x + 1 are linearly independent or linearly dependent.
(d) Does the polynomial q(x) = x3 lie in their span? If so produce a linear combination that adds up to q(x).

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