Objectives
- Understand the definition of and what it means to use to label points on a geometric object.
- Pictures: solutions of systems of linear equations, parameterized solution sets.
- Vocabulary words: consistent, inconsistent, solution set.
Section1.1Systems of Linear Equations¶ permalink
In the first half of this textbook, we are primarily concerned with understanding the solutions of systems of linear equations.
Definition
An equation in the variables is called linear if both sides of the equation are a sum of (constant) multiples of plus an optional constant.
For instance,
are both linear equations, but neither of
are linear. The first equation fails to be linear because of the product $yz$ and the second equation fails to be linear because of the non-linear functions and
As a convention, we generally move the terms with variables 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
(1.1.1)
The curly brace here is optional, it is just there to emphasize that we are considering all three of these equations at once.
Definition(Solution sets)
- A solution of a system of equations is a list of numbers which when substituted for the variables makes 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.
A system of linear equations need not have a solution. For example, since the following system of equations has no solution:
In this case, the solution set is empty. As this is a rather important property of a system of equations, it has its own name.
Definition
A system of equations is called inconsistent if it has no solutions. If there is at least one solution, then the system is called consistent otherwise.
A solution of a system of equations in variables is a list of numbers. For example, is a solution of (1.1.1). As we will be studying solutions of systems of equations throughout this text, now is a good time to fix our notions regarding lists of numbers.
Subsection1.1.1Line, Plane, Space, Etc.¶ permalink
We use to denote the set of all real numbers, i.e., the number line. This contains numbers like
Definition
Let be a positive whole number. We define
An -tuple of real numbers is called a point of
We will draw pictures of in a moment, but keep in mind that this is the definition. For example, and are points of
Example(The number line)
When is just the real numbers. Geometrically, this is the number line.
Example(The Euclidean plane)
When we may geometrically interpret as the -plane. Every point in corresponds to a geometric point on the plane, which is to say that every point can be represented by an ordered pair of real numbers, namely, its - and -coordinates.
Example(3-Space)
When we may geometrically interpret as the space we (appear to) live in. We can do so because every point in space can be represented by an ordered triple of real numebrs, namely, its -, -, and -coordinates.
So what is or or These are harder to visualize. Instead of trying to see these spaces, we fall back on their mathematical definition: is the set of all ordered -tuples of real numbers These are still “geometric” spaces from the point of view of mathematics, even if we may have a harder time imagining that geometry. By combining our intution from dimensions one, two, and three with the methods of algebra, we are still able to fruitfully do geometry in higher dimensions.
Subsection1.1.2Pictures 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.
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 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 equations are the points that lie on both planes. We can see in the picture below that the planes intersect in a line. Hence this system has infinitely many solutions.
Subsection1.1.3Parametric 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 describe a systematic way of doing this in Section 1.3; for now we simply present 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 3-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.