Chapter 2

Systems of Linear EquationsThis part of the book is about linear algebra itself, as a part of algebra. Some geometric andanalytic concepts do creep in, but it is primarily about algebra. It involves general fieldsand has very little to do with limits and completeness although some geometry is included,but not much. Numbers are elements of a field.

2.1 Elementary OperationsIn this chapter, the main interest is in fields of scalars consisting of R or C, but everythingis applied to arbitrary fields. Consider the following example.

Example 2.1.1 Find x and y such that

x+ y = 7 and 2x− y = 8. (2.1)

The set of ordered pairs, (x,y) which solve both equations is called the solution set.

You can verify that (x,y) = (5,2) is a solution to the above system. The interestingquestion is this: If you were not given this information to verify, how could you determinethe solution? You can do this by using the following basic operations on the equations,none of which change the set of solutions of the system of equations.

Definition 2.1.2 Elementary operations are those operations consisting of the following.

1. Interchange the order in which the equations are listed.

2. Multiply any equation by a nonzero number.

3. Replace any equation with itself added to a multiple of another equation.

Example 2.1.3 To illustrate the third of these operations on this particular system, con-sider the following.

x+ y = 72x− y = 8

The system has the same solution set as the system

x+ y = 7−3y =−6

.

To obtain the second system, take the second equation of the first system and add −2 timesthe first equation to obtain −3y =−6. Now, this clearly shows that y = 2 and so it followsfrom the other equation that x+2 = 7 and so x = 5.

Of course a linear system may involve many equations and many variables. The so-lution set is still the collection of solutions to the equations. In every case, the aboveoperations of Definition 2.1.2 do not change the set of solutions to the system of linearequations.

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