The space ℂn

So far, in this book, we have only considered vectors and matrices with real entries. However, as the reader may have already remarked, there is nothing in Chapter 1 on Gaussian elimination which will not work equally well when applied to m linear equations with complex coefficients in n complex unknowns. In particular, there is nothing to prevent us considering complex row and column vectors (z1, z2, …, zn) and (z1, z2, …, zn)T with zj ∈ ℂ and complex m × n matrices A = (aij) with aij ∈ ℂ. (If we are going to make use of the complex number i, it may be better to use other suffices and talk about A = (ars).)

Exercise 5.1.1 Explain why we cannot replace ℂ by ℤ in the discussion of the previous paragraph.

However, this smooth process does not work for the geometry of Chapter 2. It is possible to develop complex geometry to mirror real geometry, but, whilst an ancient Greek mathematician would have no difficulty understanding the meaning of the theorems of Chapter 2 as they apply to the plane or three dimensional space, he or she would find the complex analogues (when they exist) incomprehensible. Leaving aside the question of the meaning of theorems of complex geometry, the reader should note that the naive translation of the definition of inner product from real to complex vectors does not work very well. (We shall give an appropriate translation in Section 8.4.)