Algebraic Combinatorics I: Association Schemes by Eiichi Bannai

By Eiichi Bannai

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By an ordered basis 20 CHAPTER 1. INTRODUCTION we mean a basis in which the vectors of the basis are listed in a specified order; to indicate that we have an ordered basis we write (u(1) , u(2) , . . , u(m) ). A spanning set S of V is a minimal spanning set of V provided that each set of vectors obtained from S by removing a vector is not a spanning set for V . A linearly independent set S of vectors of V is a maximal linearly independent set of vectors of V provided that for each vector w of V that is not in S, S ∪ {w} is linearly dependent (when this happens, w must be a linear combination of the vectors in S).

Verification of the rest of the field properties is now routine. ✷ As an example, let m = 7. Then Z7 is a field with 2·4 =1 3·5 =1 6·6 =1 so that 2−1 = 4 and 4−1 = 2; so that 3−1 = 5 and 5−1 = 3; so that 6−1 = 6. Two fields F and F ′ are isomorphic provided there is a bijection φ : F → F ′ that preserves both addition and multiplication: φ(a + b) = φ(a) + φ(b), and φ(a · b) = φ(a) · φ(b). In these equations the leftmost binary operations (addition and multiplication, respectively) are those of F and the rightmost are those of F ′ .

Ii) (u + v) · w = u · w + v · w and u · (v + w) = u · v + u · w. (iii) cu · v = c(u · v) and u · cv = c¯u · v (so if c is a real scalar, u · cv = c(u · v)). (iv) u · v = v · u (so if u and v are real vectors. u · v = v · u) (v) (Cauchy–Schwarz inequality) |u · v| ≤ ||u||||v|| with equality if and only if u and v are linearly dependent. ✷ Let u and v be nonzero vectors in ℜn . By the Cauchy–Schwarz inequality, u·v ≤ 1. −1 ≤ ||u||||v| Hence there is an angle θ with 0 ≤ θ ≤ π such that cos θ = u·v .

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