
5: Span and Bases


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.

• 5.1: Linear Span
The linear span (or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set. The linear span of a set of vectors is therefore a vector space.
• 5.2: Linear Independence
We are now going to define the notion of linear independence of a list of vectors. This concept will be extremely important in the sections that follow, and especially when we introduce bases and the dimension of a vector space.
• 5.3: Bases
A basis of a finite-dimensional vector space is a spanning list that is also linearly independent. We will see that all bases for finite-dimensional vector spaces have the same length. This length will then be called the dimension of our vector space.
• 5.4: Dimension
We now come to the important definition of the dimension of a finite-dimensional vector space.
• 5.E: Exercises for Chapter 5

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