Abstract vector spaces
5.1 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 in the discussion of the previous paragraph ...