Linear Algebra

354 Chapter Five. Similaritythe scalars be complex so the first section is a quick review of complex numbers.In this book, our approach is to shift to this more general context of takingscalars to be complex only for the pragmatic reason that we must do so nowin order to move forward. However, the idea of doing vector spaces by takingscalars from a structure other than the real numbers is an interesting and usefulone. Delightful presentations that take this approach from the start are in[Halmos] and [Hoffman & Kunze].I.1 Review of Factoring and Complex NumbersThis subsection is a review only and we take the main results as known.For proofs, see [Birkhoff & MacLane] or [Ebbinghaus].We consider a polynomial p(x) = c n x n + · · · + c 1 x + c 0 with leadingcoefficient c n ≠ 0. The degree of the polynomial is n if n 1. If n = 0 thenp is a constant polynomial p(x) = c 0 . Constant polynomials that are not thezero polynomial, c 0 ≠ 0, have degree zero. We define the zero polynomial tohave degree −∞.Just as integers have a division operation — e.g., ‘4 goes 5 times into 21 withremainder 1’ — so do polynomials.1.1 Theorem (Division Theorem for Polynomials) Let c(x) be a polynomial. If m(x)is a non-zero polynomial then there are quotient and remainder polynomialsq(x) and r(x) such thatc(x) = m(x) · q(x) + r(x)where the degree of r(x) is strictly less than the degree of m(x).1.2 Remark Defining the degree of the zero polynomial to be −∞, which mostbut not all authors do, allows the equation degree(fg) = degree(f) + degree(g)to hold for all polynomial functions f and g.The point of the integer division statement ‘4 goes 5 times into 21 withremainder 1’ is that the remainder is less than 4 — while 4 goes 5 times, it doesnot go 6 times. In the same way, the point of the polynomial division statementis its final clause.1.3 Example If c(x) = 2x 3 − 3x 2 + 4x and m(x) = x 2 + 1 then q(x) = 2x − 3 andr(x) = 2x + 3. Note that r(x) has a lower degree than m(x).1.4 Corollary The remainder when c(x) is divided by x − λ is the constantpolynomial r(x) = c(λ).