Intuition probably tells you that the plane $$\mathbb{R}^2$$ is of dimension two and that the space we live in $$\mathbb{R}^3$$ is of dimension three. You have probably also learned in physics that space-time has dimension four and that string theories are models that can live in ten dimensions. In this chapter we will give a mathematical definition of the dimension of a vector space. For this we will first need the notions of linear span, linear independence, and the basis of a vector space.