# 5 Linear subspaces and spans

When we defined our vectors in Euclidean space, we saw how to perform the operations of addition and scalar multiplication. A common theme of linear algebra is to study first and foremost these two key operations and try to give them intuitive or geometric meaning. Having seen how these operations are defined on \(\mathbb{R}^n\), it is natural to next turn to thinking about the behaviour of subsets of \(\mathbb{R}^n\), and consider the interaction between subsets and these operations.

Therefore, in this chapter we want to study the following two closely related questions:

Which type of subsets of \(\mathbb{R}^n\) (or \(\mathbb{C}^n\), or a vector space in general) stay invariant under these two operations?

Which type of subsets of \(\mathbb{R}^n\) (or \(\mathbb{C}^n\), or a vector space in general) can be generated by using these two operations?

Again, we will focus on \(\mathbb{R}^n\) and \(\mathbb{C}^n\) in this chapter, but these results can be generalised to other vector spaces as well.