Practice Makes Perfect Linear Algebra

McGraw-Hill Companies, Inc.

Contents

Chapter One

In this chapter, you will learn:

* Systems of linear equations

* General systems of linear equations

* Matrices

* Row transformations and equivalence of matrices

* Row-echelon form

* Homogeneous systems

This chapter provides some basic terminology associated with systems of linear equations and matrices and presents methods for solving linear systems. As you will see, systems of linear equations and matrices go hand-in-hand.

Systems of linear equations

A linear equation is an equation of the form a₁x₁ + a₂x₂ + ... a_nx_n = c₁ where there are n unknowns (variables), x₁, x₂, ..., x_n, and a₁, a₂, ..., a_n and c₁ are known constants. An equation is linear if all the variables occur to the first power only and if there are no products or quotients of variables. A solution of a linear equation a₁x₁ + a₂x₂ + ... + a_nx_n = c₁ in n unknowns is an ordered list of n numbers s₁, s₂, ..., s_n that satisfy the equation when x₁ = s₁, x₂, = s₂, ..., x_n = s_n. In linear algebra you will need to simultaneously solve linear equations in several variables. You likely have seen two basic algebraic methods of solving simultaneous equations. An example of each method is shown for a quick review.

The equation of a linear function in two variables has several forms but for the purposes of this chapter, the form ax + by = c is preferred. You know from elementary algebra that the graph of such a linear equation is a line. When you encounter two linear equations, the basic question underlying a simultaneous solution is "What are the coordinates of the point of intersection, if any, of the two lines that are the graphs of the two equations?" This question has three possible answers. If the two lines do not intersect, there is no solution; if the two lines intersect, there is only one solution, an ordered pair (x, y); and if the two lines are equivalent versions of the same line, then there are infinitely many solutions: an infinite set of ordered pairs. This same idea carries over to systems of m equations in n unknowns. That is,

Given a system of m equations and n unknowns, only one of the following three possibilities for a solution to the system occurs:

1. There is no solution.

2. There is exactly one solution.

3. There are infinitely many solutions.

If there is no solution to the system, the system is inconsistent. If there is at least one solution to the system, the system is consistent.

When you are solving two linear equations in two variables (a 2 x 2 system), an example of the standard form of writing them together is

2x - y = 1

x + y = 2

Figures 1.1, 1.2, and 1.3 graphically show the three possible outcomes for a 2 x 2 system of equations.

You solve the 2 x 2 system when you answer the question: "What are the coordinates of the point of intersection, if any, of the two lines that are the graphs of the two equations?"

Method 1: Substitution. This method requires you to solve one equation for one of the variables in terms of the other variable, and then use substitution to solve the system.

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SOLUTION Solve the first equation for y in terms of x:

2x = y = 0

2x = y

Substitute 2x for y into the second equation and solve for x:

x + (2x) = 3

x = 1

Substitute 1 for x into the second equation and solve for y:

1 + y = 3

y = 2

The solution is x = 1 and y = 2. That is, the two lines intersect at the point (1, 2).

Method 2: Elimination of a variable. This method involves multiplying the equations by constants to create opposites as coefficients of one variable so that it can be eliminated by adding the two equations.

PROBLEM Solve the system.

5x + 2y = 3

2x + 3y = -1

SOLUTION To eliminate y, multiply the first equation by 3 and the second equation by -2:

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Add the resulting two equations and solve for x:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Substitute 1 for x into one of the original equations and solve for y:

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The solution is x = 1 and y = -1. That is, the two lines intersect at the point (1, -1).

You can use the two methods exemplified above to solve systems of several linear equations in several variables, but linear algebra is a study of more powerful ways of solving systems of equations. Solving systems of equations is elevated to answering more sophisticated questions than "Where do lines intersect?" Moreover, solving such systems is basic to the study of linear algebra. Chapter 2 presents a discussion of the algebra of matrices and methods of solving simultaneous equations using that algebra. As you might expect, the solution(s) of 2 x 2 systems are ordered pairs; 3 x 3 systems, ordered triples; 4 x 4 systems, ordered quadruples; and, in general, n x n systems, ordered n-tuples.

General systems of linear equations

The simple 1 x 2 linear system ax + by = c with nonzero coefficients has infinitely many solutions whose graph is a straight line. The solutions can be written as (x, y), where [MATHEMATICAL NOT REPRODUCIBLE IN ASCII]. The variable x is a free variable, meaning its value is unrestricted. The variable y is dependent on x because substituting any value for x will produce a corresponding value for y. By letting x = t, you can represent the solutions in parametric form as x = t, [MATHEMATICAL NOT REPRODUCIBLE IN ASCII], where the arbitrary number t is called a parameter. Similar parametric substitutions are made if you have several free variables.

PROBLEM Solve the following system: -2x + 5y = 4z = 0

SOLUTION Solve for one variable in terms of the other two:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII], where y and z are free variables. In parametric form, the solution is [MATHEMATICAL NOT REPRODUCIBLE IN ASCII], y = s and z = t, where s and t are arbitrary.

A system of linear equations can have any number of equations and any number of variables. The solution to many such systems...