In this chapter we study vector spaces and their basic properties and structures. We start by stating the definition and a discussion of the examples of vector spaces. We next introduce the notions of subspaces, linear dependence, bases, coordinates, and dimensionality. We then consider dual spaces, direct sums, and quotient spaces. Finally we cover normed vector spaces.

Vector spaces

A vector space is a non-empty set consisting of elements called vectors which can be added and multiplied by some quantities called scalars. In this section, we start with a study of vector spaces.

1.1.1 Fields

The scalars to operate on vectors in a vector space are required to form a field, which may be denoted by F, where two operations, usually called addition, denoted by ‘+’, and multiplication, denoted by ‘·’ or omitted, over F are performed between scalars, such that the following axioms are satisfied.

1. (Closure) If a, b ∈ F, then a + b ∈ F and ab ∈ F.

2. (Commutativity) For a, b ∈ F, there hold a + b = b + a and ab = ba.

3. (Associativity) For a, b, c ∈ F, there hold (a + b) + c = a + (b + c) and a(bc) = (ab)c.

4. (Distributivity) For a, b, c ∈ F, there hold a(b + c) = ab + ac.

5. (Existence of zero) There is a scalar, called zero, denoted by 0, such that a + 0 = a for any a ∈ F.

6. (Existence of unity) There is a scalar different from zero, called one, denoted by 1, such that 1a = a for any a ∈ F.

7. (Existence of additive inverse) For any a ∈ F, there is a scalar, denoted by −a or (−a), such that a + (−a) = 0.

8. (Existence of multiplicative inverse) For any a ∈ F \ {0}, there is a scalar, denoted by a−1, such that aa−1 = 1.