Introduction to Linear Algebra

The concept of a vector is basic for the study of functions of several variables. It provides geometric motivation for everything that follows. Hence the properties of vectors, both algebraic and geometric, will be discussed in full.

You have met linear equations in elementary school. Linear equations are simply equations like 2x+y+ z= 1, 5x—y+7z=0. You have learned to solve such equations by the successive elimination of the variables. In this chapter, we shall review the theory of such equations, dealing with equations in n variables, and interpreting our results from the point of view of vectors. Several geometric interpretations for the solutions of the equations will be given.

As usual, a collection of objects will be called a set. A member of the collection is also called an element of the set. It is useful in practice to use short symbols to denote certain sets. For instance we denote by R the set of all numbers. To say that “x is a number” or that “x is an element of R ⁿ amounts to the same thing. The set of n-tuples of numbers will be denoted by W. Thus “X is an element of R ⁿ and ”X is an n-tuple“ mean the same thing. Instead of saying that u is an element of a set S, we shall also frequently say that u lies in S and we write u E S. If S and S’ are two sets, and if every element of S’ is an element of S, then we say that S’ is a subset of S. Thus the set of rational numbers is a subset of the set of (real) numbers. To say that S is a subset of S’ is to say that S is part of S’. To denote the fact that S is a subset of S’, we write S S’.

We shall first define the general notion of a mapping, which generalizes the notion of a function. Among mappings, the linear mappings are the most important. A good deal of mathematics is devoted to reducing questions concerning arbitrary mappings to linear mappings. For one thing, they are interesting in themselves, and many mappings are linear. On the other hand, it is often possible to approximate an arbitrary mapping by a linear one, whose study is much easier than the study of the original mapping. This is done in the calculus of several variables.

LetU, V, Wbe sets. LetF:U→VandG:V→Wbe mappings. Then we can form the composite mapping fromUintoWdenoted byG○F. It is by definition the mapping such that (G○F)(u) =G(F(u)) for alluinU.

LetVbe a vector space. Ascalar productonVis an association which to any pair of elements (v, w) ofVassociates a number, denoted by <v, w>, satisfying the following properties:

We have worked with vectors for some time, and we have often felt the need of a method to determine when vectors are linearly independent. Up to now, the only method available to us was to solve a system of linear equations by the elimination method. In this chapter, we shall exhibit a very efficient computational method to solve linear equations, and determine when vectors are linearly independent.

This chapter gives the basic elementary properties of eigenvectors and eigenvalues. We get an application of determinants in computing the characteristic polynomial. In §3, we also get an elegant mixture of calculus and linear algebra by relating eigenvectors with the problem of finding the maximum and minimum of a quadratic function on the sphere. Most students taking linear algebra will have had some calculus, but the proof using complex numbers instead of the maximum principle can be used to get real eigenvalues of a symmetric matrix if the calculus has to be avoided. Basic properties of the complex numbers will be recalled in an appendix.