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REVIEW OF PREREQUISITES

Part 1 of this chapter summarizes some pre-calculus prerequisites for this book-facts about real numbers, rectangular coordinates, complex numbers, and mathematical induction. Part 2 does the same for calculus prerequisites. Chapters 1 and 2, which deal with vector algebra and its applications to analytic geometry, do not require calculus as a prerequisite. These two chapters provide motivation and concrete examples to illustrate the abstract treatment of linear algebra that begins with Chapter 3. In Chapters 3 through 7, calculus concepts occur only occasionally in some illustrative examples, or in some exercises; these are clearly identified and can be omitted or postponed without disrupting the continuity of the text.

Although calculus and linear algebra are independent subjects, some of the most striking applications of linear algebra involve calculus concepts-integrals, derivatives, and infinite series. Familiarity with one-variable calculus is essential to understand these applications, especially those referring to differential equations presented in the last three chapters. At the same time, the use of linear algebra places some aspects of differential equations in a natural setting and helps increase understanding.

Part 1. Pre-calculus Prerequisites

0.1 Real numbers as points on a line

Real numbers can be represented geometrically as points on a straight line. A point is selected to represent 0 and another, to the right of 0, to represent 1, as illustrated in Figure 0.1. This choice determines the scale, or unit of measure. If one adopts an appropriate set of axioms for Euclidean geometry, then each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number. For this reason, the line is usually called the real line or the real axis. We often speak of the point x rather than the point corresponding to the real number x. The set of all real numbers is denoted by R.

FIGURE 0.1 Real numbers represented geometrically on a line.

If x < y, point x lies to the left of y as shown in Figure 0.1. Each positive real number x lies at a distance x to the right of zero. A negative real number x is represented by a point located at a distance |x| to the left of zero.

0.2 Pairs of real numbers as points in a plane

Points in a plane can be represented by pairs of real numbers. Two perpendicular reference lines in the plane are chosen, a horizontal x axis and a vertical y axis. Their point of intersection, denoted by 0, is called the origin. On the x axis a convenient point is chosen to the right of 0 to represent 1; its distance from 0 is called the unit distance. Vertical distances along the y axis are usually measured with the same unit distance. Each point in the plane is assigned a pair of numbers, called its coordinates, which tell us how to locate the point. Figure 0.2 illustrates some examples. The point with coordinates (3, 2) lies three units to the right of the y axis and two units above the x axis. The number 3 is called the x coordinate or abscissa of the point, and 2 is its y coordinate or ordinate. Points to the left of the y axis have a negative abscissa; those below the x axis have a negative ordinate. The coordinates of a point, as just defined, are called its Cartesian coordinates in honor of René Descartes (1596-1650), one of the founders of analytic geometry.

When a pair of numbers is used to represent a point, we agree that the abscissa is written first, the ordinate second. For this reason, the pair (a, b) is referred to as an ordered pair: the first entry is a, the second is b. Two ordered pairs (a, b) and (c, d) represent the same point if and only if we have a = c and b = d. Points (a, b) with both a and b positive are said to lie in the first quadrant; those with a > 0 and b > 0 are in the second quadrant, those with a > 0 and b > 0 are in the third quadrant, and those with a > 0 and b > 0 are in the fourth quadrant. Figure 0.2 shows one point in each quadrant.

The procedure for locating points in space is analogous. We take three mutually perpendicular lines in space intersecting at a point (the origin). These lines determine three mutually perpendicular planes, and each point in space can be completely described by specifying, with appropriate regard for signs, the distances from these planes. We shall discuss three-dimensional Cartesian coordinates in a later chapter; for the present we confine our attention to the two-dimensional case.

FIGURE 0.2 Points in the plane represented by pairs of real numbers.

FIGURE 0.3 The circle represented by the Cartesian equation x2 + y2 = r.

A geometric figure, such as a curve in the plane, is a collection of points satisfying one or more special conditions. By expressing these conditions in terms of the coordinates x and y we obtain one or more relations (equations or inequalitites) that characterize the figure in question. For example, consider a circle of radius r with its center at the origin, as shown in Figure 0.3.

Let (x, y) denote the coordinates of an arbitrary point P on this circle. The line segment OP is the hypotenuse of a right triangle whose legs have lengths |x| and |y| and, hence, by the theorem of Pythagoras, we have

This equation, called a Cartesian equation of the circle, is satisfied by all points (x, y) on the circle and by no others, so the equation completely characterizes the circle. Points inside the circle satisfy the inequality x2 + y2 < r2, while those outside satisfy x2 + y2 > r2. This example illustrates how analytic geometry is used to reduce geometrical statements about points to algebraic relations about real numbers.

0.3 Polar coordinates

Points in a plane can also be located by using polar coordinates. This is done as follows. Let P be a point distinct from the origin. Suppose the line segment joining the origin to P has length r > 0 and makes an angle of ? radians with the positive x axis, as shown by the example in Figure 0.4. The two numbers r and ? are called polar coordinates of P. They are related to the rectangular coordinates x and y by the equations

(0.1)

FIGURE 0.4 Polar coordinates.

The positive number r is called the radial distance of P, and ? is called a polar angle. We say a polar angle rather than the polar angle because if ? satisfies (0.1) so does ? + 2np for any integer n. We agree to call all pairs of real numbers (r, ?) polar coordinates of P if they satisfy (0.1) with r > 0.

The radial distance r is uniquely determined by x and y: , but the polar angle ? is determined only up to integer multiples of 2p.

When P is the origin, Eqs. (0.1) are satisfied with r = 0 and any ?. For this reason, we assign the radial distance r = 0 to the origin, and we agree that any real ? may be used as a polar angle.

Some curves are described more simply with polar coordinates rather than rectangular coordinates. For example, a circle of radius 2 with center at the origin has Cartesian equation x2 + y2 = 4. In polar coordinates the same circle is described by the simpler equation r = 2. The interior of the circle is described by the inequality r > 2, the exterior by r > 2.

0.4 Complex numbers

The quadratic equation x2 + 1 = 0 has no solution in the real-number system because there is no real number whose square is negative. New types of numbers, called complex numbers, have been introduced to provide solutions to such equations.

As early as the 16th century, a symbol was introduced to provide solutions of the quadratic equation x2 + 1 = 0. This symbol, later denoted by the letter i, was regarded as a fictitious or imaginary number, which could be manipulated algebraically like an ordinary real number, except that its square was -1. Thus, for example, the quadratic polynomial x2 + 1 was factored by writing

and the solutions of the equation x2 + 1 =0 were exhibited as x = ±i, without any concern regarding the meaning or validity of such formulas. Expressions such as 2 + 3i were called complex numbers, and they were used in a purely formal way for nearly 300 years before they were described in a manner that would be considered...