ABSTRACT
In this chapter the general notion of a vector space is presented. The reader will be presented with the four classic vector spaces: Tuples & Matrices, which the reader has already seen, and Polynomials & Functions, which will be introduced in the first section of this chapter. There will be an increase in abstractness in the text at this point, but this is to be expected and before long overcome. In Section 3.1 a vector space is defined with the four classic examples presented among others. In Section 3.2 a subspace is defined with many examples in the context of the four classic vector spaces. The reader is shown methods for proving or disproving that a particular set of vectors is a subspace. In Section 3.3 linear independence is introduced and concrete methods for verifying linear independence/dependence are given for the four classic vector spaces. In Section 3.4 a very important subspace is defined called the span of a set of vectors. Concrete methods are given for computing the span of a set of vectors. In Section 3.5 the notions of basis and dimension are introduced. In Section 3.6 row space, column space and null space of a matrix are defined and some loose ends are tied up from earlier sections of this chapter. Finally, in Section 3.7, some counting arguments using dimension theorems are applied to a number of examples.
