Linear spaces

In essence vector spaces are quite simple mathematical objects. We can distinguish between so called ‘real vector spaces’ and the maybe less used ‘complex vector spaces’. The former uses real numbers, while the latter uses complex numbers. For the novice reader, many textbooks will give a definition of such a space and they very often will go into more detail (such as on covering the notion of subspaces; product of spaces and span). Hubbard and Hubbard [1]; Kreyszig [2] and Ha [3] are good examples.

The real vector space V is characterized by i) commutativity in addition and hence for two vectors a, b ∈ V: a + b = b + a; ii) associativity in addition and hence for vectors a, b, c ∈ V:(a + b) + c = a + (b + c); iii) the existence of a vector 0 ∈ V such that: a + 0 = a; iv) for each vector a ∈ V there exists an additive inverse, −a :(−a) + a = 0. Moreover, for real scalars a, β ∈ R and vectors a, b ∈ V, the following simple laws need to be satisfied: i) 1a = a; ii) (α + β) a = αa + βa (distributivity towards scalars); iii) α (a + b) = αa + αb (distributivity towards vectors): iv) α (βa) = (αβ) a (associativity). The complex vector space is obtained by just swapping the real scalars for complex scalars.