1.16. The Vector Space Cn of n-Tuples of Complex Numbers The set of all n-tuples of complex numbers is denoted by Cn. By replacing Rn with Cn in the definition of section 1.2, we can turn Cn into a vector space. Note that either R or C can be used as scalars, thus giving rise to a complex vector space defined on the real or complex number field, respectively. Theorem 1.1 of section 1.2 is easily proved for Cn. In order to preserve the positive property of the dot product necessary for its relation to the norm, its definition must be modified as follows. Definition (Dot Product) Let A a1, , an and B b1, , bn be 2 vectors in Cn. The dot product A B is defined as n A B ak bk k 1 where bk is the complex conjugate of bk . n Note that it is more common in physics to use the definition A B ak bk . k 1 Theorem 1.11 For all A, B, C Cn and scalars α, we have 1. A B B A 2. A B C A B A C 3. A B A B A B A B 4. A A 0 where equal sign applies only if A 0 . Proof of the theorem follows directly from the definition and are left as exercise. Cauchy-Schwarz Inequality A B A A B B 2 Definition (Norm) A A A (1.14) Note that theorems 1.4-5 of section 1.6 also hold for Cn.