1.4 Vector spaces

To finish this chapter, let’s return to our vectors in \(\mathbb{R}^n\) and summarise the properties of how these vectors interact with each other and with scalars in \(\mathbb{R}\).

We have that:

\(u+v=v+u\) for all \(u,v \in \mathbb{R}^n\) (vector addition is commutative).
\((u+v)+w=u+(v+w)\) for all \(u,v,w \in \mathbb{R}^n\) (vector addition is associative).
there exists an element \(0 \in \mathbb{R}^n\) such that \(v+0=v\) for all \(v\in \mathbb{R}^n\) (\(\mathbb{R}^n\) contains an additive identity).
for every \(v\in \mathbb{R}^n\) there exists an element \(w\in \mathbb{R}^n\) such that \(v+w =0\) (every element has an additive inverse, namely \(w=-v\)).
\(\lambda (u+v)=\lambda u+ \lambda v\) for all \(\lambda \in \mathbb{R}\) and all \(u, v \in \mathbb{R}^n\) (multiplication is distributive over addition).
\((\lambda + \mu)v = \lambda v + \mu v\) for all \(\lambda, \mu \in \mathbb{R}\) and all \(v \in \mathbb{R}^n\).
\(\lambda(\mu u)=(\lambda \mu) v\) for all \(\lambda, \mu \in \mathbb{R}\) and all \(v \in \mathbb{R}^n\).
\(1v=v\) for all \(v \in \mathbb{R}^n\).

Note that the first four properties deal only with how the vectors interact with each other, and the last four deal with how our vectors and scalars interact with each other. It is straightforward to check that all of these properties follow from our definitions of vector addition and scalar multiplication in Definition 1.2.

This combination of properties means that our vectors and scalars interact with each other in a ‘nice’ way. It is then natural to ask whether we can find other examples of sets \(V\) where we define some type of addition and scalar multiplication in a way that we satisfy these same properties. This leads us to define a vector space.

Definition 1.20: (Vector space)

A vector space is a set \(V\) together with an underlying set of scalars¹ \(\mathbb{F}\) with operations of addition, which takes vectors \(u,v \in V\) and gives an element \(u+v\in V\), and scalar multiplication, which takes a scalar \(\lambda\in \mathbb{F}\) and vector \(v \in V\) and gives an element \(\lambda v\in V\), such that:

\(u+v=v+u\) for all \(u,v \in V\).
\((u+v)+w=u+(v+w)\) for all \(u,v,w \in V\).
there exists an element \(0 \in V\) such that \(v+0=v\) for all \(v\in V\).
for every \(v\in V\) there exists an element \(w\in V\) such that \(v+w =0\).
\(\lambda (u+v)=\lambda u+ \lambda v\) for all scalars \(\lambda\in \mathbb{F}\) and all \(u, v \in V\).
\((\lambda + \mu)v = \lambda v + \mu v\) for all scalars \(\lambda\) and \(\mu\in \mathbb{F}\) and all \(v \in V\).
\(\lambda(\mu v)=(\lambda \mu) v\) for all scalars \(\lambda\) and \(\mu\in \mathbb{F}\) and all \(v \in V\).
there exists \(1\in \mathbb{F}\) such that \(1v=v\) for all \(v \in V\).

For now, we think of our set of scalars as the real numbers, and refer to \(V\) as a vector space over \(\mathbb{R}\), or a real vector space. Later in the course we will explore the requirements for our set of scalars in more detail. Our key example of a vector space is \(\mathbb{R}^n\), and this will be the focus for much of the course. However, many of the definitions and theorems that we will see in the context of \(\mathbb{R}^n\) can be generalised to any vector space. We will also see another core example of a vector space in our next chapter, where we will explore the complex numbers.

Formally, we need our set of scalars to be an algebraic object known as a field. This is a set with addition and multiplication that are associative, commutative and distributive; we need an additive and multiplicative identity; every element needs an additve inverse; and every non-zero element needs a multiplicative inverse. For now, we will just think of this as \(\mathbb{R}\).↩︎