# 5: Span and Bases

- Page ID
- 288

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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.

## Contributors

- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis

Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.