In the previous chapter we looked at vectors in Euclidean n-space and while in  and  we thought of vectors as directed line segments.  A vector however, is a much more general concept and it doesn’t necessarily have to represent a directed line segment in  or .  Nor does a vector have to represent the vectors we looked at in .  As we’ll soon see a vector can be a matrix or a function and that’s only a couple of possibilities for vectors.  With all that said a good many of our examples will be examples from  since that is a setting that most people are familiar with and/or can visualize.  We will however try to include the occasional example that does not lie in .

The main idea of study in this chapter is that of a vector space.  A vector space is nothing more than a collection of vectors (whatever those now are…) that satisfies a set of axioms.  Once we get the general definition of a vector and a vector space out of the way we’ll look at many of the important ideas that come with vector spaces.  Towards the end of the chapter we’ll take a quick look at inner product spaces.

Here is a listing of the topics in this chapter.

Vector Spaces  In this section we’ll formally define vectors and vector spaces.

Subspaces  Here we will be looking at vector spaces that live inside of other vector spaces.

Span  The concept of the span of a set of vectors will be investigated in this section.

Linear Independence  Here we will take a look at what it means for a set of vectors to be linearly independent or linearly dependent.

Basis and Dimension  We’ll be looking at the idea of a set of basis vectors and the dimension of a vector space.

Change of Basis  In this section we will see how to change the set of basis vectors for a vector space.

Fundamental Subspaces  Here we will take a look at some of the fundamental subspaces of a matrix, including the row space, column space and null space.

Inner Product Spaces  We will be looking at a special kind of vector spaces in this section as well as define the inner product.

Orthonormal Basis  In this section we will develop and use the Gram-Schmidt process for constructing an orthogonal/orthonormal basis for an inner product space.

Least Squares  In this section we’ll take a look at an application of some of the ideas that we will be discussing in this chapter.

QR-Decomposition  Here we will take a look at the QR-Decomposition for a matrix and how it can be used in the least squares process.

Orthogonal Matrices  We will take a look at a special kind of matrix, the orthogonal matrix, in this section.