Linear Systems of Equations

Goals: The fundamental problem of Linear Algebra.
The Method of Elimination.
The Augmented Matrix form, (A | b), for a system of linear equations.

I. Linear Systems of Equations

The fundamental problem of Linear Algebra is how to solve systems of linear equations. Problems whose solution required solving systems of linear equations appear in many of the ancient texts on mathematics. Here are two examples.

The Chiu ChangSuan Shu
or Nine Chapters on the Mathematical Art, is an first century Chinese text of mathematics problems. The following problem comes from the text:

"There are three classes of grain, of which three bundles of the first class, two of the second, and one of the third make 39 measures. Two of the first, three of the second and, one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of grain are contained in one bundle of each class?"

The Babylonians also studied problems which lead to simultaneous linear equations. A clay tablet dating from around 300 BC contains the following problem:

"There are two fields whose total area is 1800 square yards. One produces grain at the rate of 2/3 of a bushel per square yard while the other produces grain at the rate of 1/2 a bushel per square yard. If the total yield is 1100 bushels, what is the size of each field."

Rather interesting is the fact that both problems are related to grain production.

Solving a Nonlinear System of Equations

In precalculus times we learned how to solve a system of two equations in two unknowns using several techniques. A solution is a set of numbers (scalars) that satisfy the system of equations.

Solve the system:

An Inventory of Techniques:

a) The method of Substitution

b) The method of Elimination

c) Solve Graphically

d) Solve Numerically

e) Guess & Check

The previous system of equations is nonlinear. For what follows we are interested only in Linear Systems of Equations. At first this may seem like a major restriction, but "Well over 75% of all mathematical problems...involve solving a linear system at some stage." [Leon p. 1] Consequently, Linear Algebra provides the next major set of problem solving tools after the calculus for the scientist and mathematician.

Definition: A System of Linear Equations

A linear equation in n unknowns is an equation of the form:

where the and are real or complex numbers and the are variables.

Akamai: Linear means that the exponents of all the variables are 1.
Don't get thrown off by the subscripts.

A system of m linear equations in n unknowns has the general form:

.
:

In this case we have more than one equation, so the subscripts of the coefficients contain two integers referring to the row and column. Always think row-column!

II. The Geometry of 2x2 and 3x3 Systems of Equations

2x2 Systems

The general form:

The graph of these two equations are lines in the plane, R².

For example, let

Try changing the 's.

The intersection point of the two lines is the solution of the system.

Write the above system as two equations
in slope-intercept form, y = mx + b.

1) Let be the dependent variable
2) Solve each equation for

What are the possible cases for solutions?

1) Lines intersect:A unique solution -- the system is Consistent.

2) Parallel Lines: No Solution -- the system is Inconsistent.

3) A single line: Infinite number of solutions.

3x3 Systems

The general form:

The graph of each of these equations is a plane in R³.

Let

(Try changing these values)

Drag to rotate!

Add N and O

What are the possibilities for solutions?

Can you tell this from the equations?

III. The Vector Interpretation of a 2 x 2 System of Linear Equations

While the geometric interpretation of a system of linear equations is a helpful tool in visualizing linear systems, there is another way of looking at systems that is also useful for a conceptual understanding of systems of linear equations, and it is the idea that a linear system can be thought of as a sum of vectors. To see this consider the 2 x 2 system:

Let

and

then the system cold be written in vector form as:

or

a vector equation where x and y are scalar multiples of
and that add up to .

Solving the system of linear equations we find,

We say that is a linear combination of and . This approach to systems of linear equations we will see when we extend the ideas of linear algebra to handle vector spaces.

Vector spaces are used in multivariable calculus, so you may have already seen them.

IV. Equivalent Systems

Definition: Equivalent Systems
Two systems of equations involving the same variables are equivalent if they have the same solution set.

When you first learned algebra, you learned how to manipulate an equation to find a solution. Each of these manipulations gave an equivalent equation, that is the solution set remained unchanged. For linear systems we are primarily concerned with only three operations, called Elementary Row Operations.

is equivalent to

is equivalent to

The system

is equivalent to the system

Since and satisfy both equations.

The Elementary Row Operations

E1. Interchange two rows
E2. Multiply a row by a nonzero number
E3. Replace a row by its sum with a multiple of another row

Each of the Elementary Row Operations preserves the solution set of the original system, that is it generates an equivalent system.

V. The Matrix of a System of Linear Equations

Although Arthur Cayley in 1855 gave the first abstract definition for the matrix, the text Nine Chapters of the Mathematical Art gives the first known example of matrix methods. The problem was solved using an array of numbers arranged on a counting board. The idea of using a matrix to represent a linear system is so natural that only a few examples are needed to make the concept clear.

Definition: Matrix (Latin matre -- mother, womb)
1. That which holds things together
2. Math -- a rectangular array of mathematical elements

The matrix of a system of linear equations is formed by writing only the coefficients of the variables and the constants in an array.

The system

is written as

VI. The Method of Elimination

Although we have several available techniques for solving systems of linear equations, we will focus on the method of elimination. The basic idea behind the method of elimination is to use the elementary row operations to eliminate all but one of the variables on a given row. The method is easy to program to run on a computer and consequently the method and it's variations are the ones currently used by the computer packages designed to solve linear systems.

Solve by Elimination:

Linear System Form

Matrix Form

In this example we will eliminate
the x variable in the second equation.

Notation:

E2: R2*2®R2

E3: R2+R1®R2

E2: R2*1/5®R2

Akamai: The Elementary Row Operations are exactly those we use when we solve a System of Linear Equations using the Method of Elimination.

Look at the example above, Method of Elimination, and identify the Elementary Row Operations used.

Solve the System using the Method of Elimination. Write a matrix representation for each step.

The System of Linear Equations

The Augmented Matrix

E3: -4*R1 + R2®R2

E3: -2*R1 + R3®R3

E2 & E3: 4*R2 + -7*R3®R3

From this last step we find z = 1. We then back-substitute to find the values for x and y. Our solution set is (2, -2, 1). The last System of Equations is equivalent to the original system.

Check the solution!

The method of elimination that we have used so far has been a bit haphazard. We begin now to look for refinements of the procedure in order to develop an efficient algorithm. This is really the problem what is the most efficient way to solve a system of linear equations?

VII. The Augmented Matrix Representation of a Linear System

The matrix representation that we introduced composed of the coefficients of the variables and a column of constants is called the augmented matrix. It is a standard practice to refer to the

coefficient matrix by using the capital letter A and the column of constants with the letter b.

Given an mxn System of Linear Equations,

:
:

The Coefficient Matrix A of the Linear System is the matrix formed only of the 's, the coefficients of the variables.

The Augmented Matrix (A | b) is

The idea of representing a System of Linear Equations by a matrix is what I refer to as The First Abstraction of Linear Algebra. We are ready for a bit of terminology, the coefficient matrix and the augmented matrix for a System of Linear Equations.

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Lecture 1 Notes