ABSTRACT
The fundamental objects of study in linear algebra are vector spaces and linear transformations. This chapter examines what a vector space is; derive additional properties of a vector space beyond those explicitly required by the axioms of the definition. In showing that a set of vectors is nonempty, one can usually observe that the set contains the zero vector. In a finite dimensional vector space, all bases have the same number of vectors. The number of vectors in a basis of a vector space is the dimension of the vector space. There are two things we want to emphasize. The first is the relationship between a vector and a basis for the vector space in which the vector lies. The second point is that we have defined a basis to be a collection of vectors that are linearly independent and span a vector space.
