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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.

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

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:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

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:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

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 involves free variables.

Here are examples of systems of linear equations.

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Matrices

A matrix is a rectangular array of elements. Matrices are used as convenient ways of organizing selected sets of data. A matrix consists of rows and columns of elements within brackets and its size (dimensions) is determined by the number of rows and columns. The notation m x n (read "m by n") is used to indicate a matrix's size where m is its number of rows and n is its number of columns. For example the following is a 2 x 3 matrix. The subscript on the brackets is a useful way of indicating a matrix's size.

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

If a matrix has only one row or one column, it is a row matrix or a column matrix. Row and column matrices are also called row and column vectors, respectively.

A matrix is a square matrix if the number of rows is equal to the number of columns. The main diagonal of a square matrix is the diagonal from the upper left element to the lower right element. An identity matrix, denoted by the letter I, is a square matrix all of whose main diagonal elements are 1s and the remaining elements are 0s. For example, [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] is the 2 x 2 identity matrix.

Note: Commonly, a matrix's main diagonal is called simply its diagonal.

A matrix all of whose elements are 0s is called a zero matrix, 0_mxn.

Two matrices are equal if and only if they are the same size and their corresponding elements are equal. For example,

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Row transformations and equivalence of matrices

Two matrices are equivalent if one is obtained from the other by using one or more of the following row transformations:

1. Multiply each element of a row by the same nonzero constant.

2. Interchange any two rows.

3. Add a nonzero constant multiple of the elements of one row to the corresponding elements of another row.

Here are convenient notations for the three types of row transformations:

1. Multiply row i by the constant c: cRi

2. Interchange rows i and j: R_i [left and right arrow] R_j

3. Add c times row i to row j: cR_i + R_j

Note: Equivalence is not to be confused with equality. The symbol ~ is used to indicate equivalence.

Column transformations are similar, but only row transformations will be used for the examples and problems.

These concepts are presented at this time because judicious use of these ideas enable you to reduce a matrix that represents a given system of equations to an equivalent matrix in which the solution to its corresponding reduced system of equations is obvious or at least simpler to solve. Systems of equations corresponding to equivalent matrix representations all have the same solution set, so the solution to the reduced system of equations is the solution to the original system as well.

For the system of equations [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] the matrix [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] is the coefficient matrix and the matrix [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] is the augmented matrix of the system. Augmented matrices for larger systems of equations are similarly constructed.

The following examples illustrate the reduction process.

PROBLEM Solve the system: [MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

SOLUTION Form the augmented matrix [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] and do the indicated row transformations:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

and [MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

The reduced equivalent system is now [MATHEMATICAL NOT REPRODUCIBLE IN ASCII], for which the solution x = 2 and y = 1 is obvious. Equivalent systems all have the same solution set, so this solution is the solution to the original system as well.

PROBLEM Solve the system: [MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

SOLUTION Form the augmented matrix and do the indicated row transformations:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

The solution is x = 5, y = 1, and z = 2. You can check the solution by substituting these values into the original equations. The calculations (5)-2(-1)+3(2) =1, (5)+3(-1)-(2) = 4, and 2(5)+ (-1)-2(-2) = 13 verify the solution.

The underlying idea behind using row transformations is to transform the coefficient matrix as closely as possible to an upper triangular matrix whose diagonal elements are all 1s. This result will then make the solution simpler to determine. This idea is formalized in the next section. (See Chapter 4 for a discussion of upper triangular matrices).

Note: Graphing calculators have built-in functions that will do row transformations. Intelligent use of these calculators will save valuable time and take the tedium out of calculations. Solving simultaneous equations with the use of a graphing calculator is detailed in Chapter 3.

Row-echelon form

A matrix is in row-echelon form if it satisfies the following three conditions.

1. If there are any rows of all zeros, then they are the lower (bottom) rows of the matrix.

2. If a row does not consist of all zeros, then its first (leftmost) nonzero entry is 1. This 1 is called a leading 1.

3. In any two successive rows, neither of which consists of all zeros, the leading 1 of the lower row is to the right of the leading 1 of the row above.

A matrix is in reduced row-echelon form if it satisfies items 1â€"3 and in addition satisfies the following fourth condition.

4. If a column contains a leading 1, then all the other entries of that column are zero.

Reducing the augmented matrix for a system to row-echelon form is called Gaussian elimination. Continuing the reduction to reduced row-echelon form is called Gauss-Jordan elimination.

The matrix [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] is in row-echelon form and the matrix [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] is in reduced row-echelon form. These forms are desired because the solutions of augmented systems are readily obtained if the system has been reduced to one of these forms.

PROBLEM Use Gauss-Jordan elimination to solve the system:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

SOLUTION Form the augmented matrix [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] and do the indicated row transformations:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Note: Read the symbol "[??]" as implies that.

Remark: This transformation process can be shortened at times by doing two things at a time, but doing more than two things at a time is discouraged due to the greater chance of errors.

PROBLEM Solve the given system.

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

SOLUTION

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Form the augmented matrix and do the indicated row transformations:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Form the augmented matrix and do the indicated row transformations:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

There are infinitely many solutions. For instance, one solution is z = -1, x = 8, y = 4.

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

There is no solution because the last row corresponds to the equation 0x + 0y + 0z = 1, which of course is not valid.

Tip: Whenever a row of zeros appears in the coefficient matrix and a nonzero element is the constant in that row, the system is inconsistent.

Form the augmented matrix and do the indicated row transformations:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Form the augmented matrix and do the indicated row transformations:

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Note: See Chapter 3 for calculator methods for solving systems of linear equations using Gauss-Jordan elimination.

Homogeneous systems

A system equations is said to be homogeneous if the constant terms are all zero. The system has the form

[MATHEMATICAL NOT REPRODUCIBLE IN ASCII]

Such systems are special because of the solution possibilities. Every homogeneous system always has a solution because x₁ = 0, x₂ = 0, x_n = 0 is a solution, called the trivial solution. Any other solution is a nontrivial solution. In fact, there are only two possibilities for solutions.

(Continues...)

Excerpted from Practice Makes Perfect Linear Algebraby Sandra Luna McCune William D. Clark Copyright © 2013 by The McGraw-Hill Companies, Inc.. Excerpted by permission of McGraw-Hill Companies, Inc.. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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