Learn how a system of linear equations corresponds to a vector equation.

Learn what is meant by a solution to a system of linear equations.

Understand the solutions to systems of linear equations geometrically in dimensions 2 and 3.

Vocabulary:system of linear equations, consistent system, inconsistent system.

A major part of linear algebra is understanding the solutions of systems of linear equations.

Definition

An equation in the unknowns is called linear if both sides of the equation are a sum of (constant) multiples of plus an optional constant.

For instance,

are linear equations, but

are not.

We will usually move the unknowns to the left side of the equation, and move the constants to the right.

A system of linear equations is a collection of several linear equations, like

(2.1.1)

Definition(Solution sets)

A solution of a system of equations is a list of numbers that make all of the equations true simultaneously.

The solution set of a system of equations is the collection of all solutions.

Solving the system means finding all solutions with formulas involving some number of parameters.

In the previous chapter, we encountered vector equations. It turns out that vector equations are simply systems of linear equations in different notation. We'll give two examples to show this is the case.

A system of linear equations need not have a solution. We have already seen an example of this in vector notation (example in Section 1.2) . Here is another example: there do not exist numbers and making the following two equations true simultaneously:

In this case, the solution set is empty, and the system is said to be inconsistent.

Definition

A system of linear equations is called inconsistent if it has no solutions. It is called consistent if it has at least one solution.

This definition exactly matches the definition for vector equations (definition in Section 1.2). A system of linear equations is consistent exactly when the associated vector equation is.

Subsection2.1.1Pictures of Solution Sets¶ permalink

Before discussing how to solve a system of linear equations below, it is helpful to see some pictures of what these solution sets look like geometrically.

One Equation in Two Variables

Consider the linear equation We can rewrite this as which defines a line in the plane: the slope is and the -intercept is

Definition(Lines)

For our purposes, a line is a ray that is straight and infinite in both directions.

One Equation in Three Variables

Consider the linear equation This is the implicit equation for a plane in space.

Definition(Planes)

A plane is a flat sheet that is infinite in all directions.

The equation defines a “-plane” in -space, and more generally, a single linear equation in variables defines an “-plane” in -space. We will make these statements precise in Section 3.4.

Two Equations in Two Variables

Now consider the system of two linear equations

Each equation individually defines a line in the plane, pictured below.

A solution to the system of both equations is a pair of numbers that makes both equations true at once. In other words, it as a point that lies on both lines simultaneously. We can see in the picture above that there is only one point where the lines intersect: therefore, this system has exactly one solution. (This solution is as the reader can verify.)

Usually, two lines in the plane will intersect in one point, but of course this is not always the case. Consider now the system of equations

These define parallel lines in the plane.

The fact that that the lines do not intersect means that the system of equations has no solution. Of course, this is easy to see algebraically: if then it is cannot also be the case that

There is one more possibility. Consider the system of equations

The second equation is a multiple of the first, so these equations define the same line in the plane.

In this case, there are infinitely many solutions of the system of equations.

Two Equations in Three Variables

Consider the system of two linear equations

Each equation individually defines a plane in space. The solutions of the system of both equations are the points that lie on both planes. We can see in the picture below that the planes intersect in a line. In particular, this system has infinitely many solutions.

In general, the solutions of a system of equations in variables is the intersection of “-planes” in -space. This is always some kind of linear space, as we will discuss in Section 3.1.

Subsection2.1.2Parametric Description of Solution Sets

According to this definition, solving a system of equations means writing down all solutions in terms of some number of parameters. We will give a systematic way of doing so in Section 2.3; for now we give parametric descriptions in the examples of the previous subsection.

Lines

Consider the linear equation of this example. In this context, we call an implicit equation of the line. We can write the same line in parametric form as follows:

This means that every point on the line has the form for some real number In this case, we call a parameter, as it parameterizes the points on the line.

Now consider the system of two linear equations

of this example. These collectively form the implicit equations for a line in (At least two equations are needed to define a line in space.) This line also has a parametric form with one parameter

Note that in each case, the parameter allows us to use to label the points on the line. However, neither line is the same as the number line indeed, every point on the first line has two coordinates, like the point and every point on the second line has three coordinates, like

Planes

Consider the linear equation of this example. This is an implicit equation of a plane in space. This plane has an equation in parametric form: we can write every point on the plane as

In this case, we need two parameters and to describe all points on the plane.

Note that the parameters allow us to use to label the points on the plane. However, this plane is not the same as the plane indeed, every point on this plane has three coordinates, like the point

When there is a unique solution, as in this example, it is not necessary to use parameters to describe the solution set.