Machine learning algorithms work with data matrices, which can be viewed as collections of row vectors or as collections of column vectors. For example, one can view the rows of an n × d data matrix D as a set of n points in a space of dimensionality d, and one can view the columns as features. These collections of row vectors and column vectors define vector spaces. In this chapter, we will introduce the basic properties of vector spaces and their connections to solving linear systems of equations. This problem is also a special case of the problem of linear regression, which is one of the fundamental building blocks of machine learning.