Matrices and Linear Systems

1.1 INTRODUCTION

We will begin by discussing two familiar problems which will serve as motivation for much of what will follow.

First of all, you are all familiar with the problem of finding the solution (or solutions) of a system of linear equations. For example the system

x - y + z = 3, 3x + 2y - z = 0, 2x + y + 2z = 3, (1.1)

can be easily shown to have the unique solution x = 1, y = - 1, z = 1. Most of the techniques you have learned for finding solutions of systems of this type become very unwieldy if the number of unknowns is large or if the coefficients are not integers. It is not uncommon today for scientists to encounter systems like (1.1) containing several thousand equations in several thousand unknowns. Even using the most efficient techniques known, a fantastic amount of arithmetic must be done to solve such a system. The development of high-speed computational machines in the last 20 years has made the solution of such problems feasible.

Using the elementary analytical geometry of three-space, one can provide a convenient and fruitful geometric interpretation for the system (1.1). Since each of the three equations represents a plane, we would normally expect the three planes to intersect in precisely one point, in this case the point with coordinates (1, - 1, 1). Our geometric point of view suggests that this will not always be the case for such systems since the following two special cases might arise:

1. Two of the planes might be parallel, in which case there would be no points common to the three planes and hence no solution of the system.

2. The planes might intersect in a line and hence there would be an infinite number of solutions.

The first of these special cases could be illustrated in the system obtained from (1.1) by replacing the third equation by 3x + 2y – z = 5; the second special case could be illustrated by replacing the third equation in (1.1) by the equation 4x + y = 3.

Let us now look at a general system like (1.1). Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where the aij and the ki are known constants and the xi are the unknowns, so that we have m equations in n unknowns. Note, by the way, the advantages of the double subscript notation in writing the general linear system (1.2). It may be difficult for you to interpret this system geometrically as we did with (1.1), but it certainly would not be inappropriate to use geometric language and insight (motivated by the special cases m, n ≤ = 3) to discuss the general linear system (1.2).

The questions we are interested in for the system (1.2) are:

1. Do solutions exist?

2. If a solution exists, is it unique?

3. How are the solutions to be found?

The second familiar problem we shall mention is that of finding the principal axes of a conic section. The curve defined by

ax2 + bxy + cy2 = d (1.3)

can always be represented by the simpler equation

a'x'2 + c'y'2 = d, (1.4)

and hence easily recognized, if we pick the x'- and yx' -coordinate system properly, that is, by rotating the x-, y -coordinate system through the proper angle. In other words, there is a coordinate system which in some sense is most natural or most convenient for investigating the curve defined by (1.3).

This problem becomes very complicated — even in three dimensions — if one uses only elementary techniques. Later on we will discover efficient ways of finding the most convenient coordinate system for investigating the quadratic function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

and more generally we will consider finding coordinate systems relative to which the discussion of other problems is simplest.

We will find it convenient to use a small amount of standard mathematical shorthand as we proceed. Learning this notation is part of the education of every mathematician. In addition the gain in efficiency will be well worth any initial inconvenience.

A set of objects can be specified by listing its members or by giving some criteria for membership. For example,

{(x, y)|x2 + y2 = 25, x ≥ 0}

is the set of all points on a circle of radius 5 with center at the origin and which lie in the right half-plane. We will commonly use this "set-builder" notation in describing sets. If a is an element of a set S, we will write a [member of] S; and if a is not in S, we will write a [not member of] S.

The symbols [there exists] and [for all] are common shorthand for "there exists" and "for all" or "for any." The term "such that" is commonly abbreviated "[member of]". Thus we would read

[for all] a [member of] S [there exists]g [member of] S [contains as member] ag = 1

as "for any a in S there exists g in S such that ag = 1."

We will use a double-shafted arrow [??] to indicate a logical implication. Thus the statement "H [??] C" can be read "if H then C," while "H [??] C" can be read as "H if and only if C."

Statements of the type H [??] C are called theorems; the statements in H are called the hypotheses and the statements in C the conclusions. In proving the theorem H [??] C it is often convenient to prove the logically equivalent assertion "not C [??] not H" which is called the contrapositive of the original theorem.

If A and B are any two sets, then the intersection of A and B is defined to be

A [intersection] B = {x|x [member of] A and x [member of] B}

and the union of A and B is defined to be

A [union] B = {x|x [member of] A or x [member of] B}

In dealing with sets and relationships between sets, it is often convenient to construct simple pictures (such as those in Fig. 1.1) which are called Venn diagrams, and which indicate graphically the relationships between the sets involved.

If two sets A and B have no elements in common, we say that they are disjoint and write A [??] B = [??], where [??] is the empty set, that is, the set with no elements. If every element of A is also an element of B, we say that A is a subset of B and write A [??] B or B [??] A. If A [??] B and [there exists] x [member of] B [contains as member] x [not member of] A, then we say that A is a proper subset of B and write A [subset] B or B [contains]...