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Section1.1What can we expect

At its heart, the subject of linear algebra is about linear equations and, more specifically, collections of linear equations. Google routinely deals with a collection of trillions of equations each of which has trillions of unknowns. We will eventually understand how to deal with that kind of complexity. To begin, however, we will look at a more familiar situation where we have a small number of equations and a small number of unknowns. In spite of its relative simplicity, this situation is rich enough to demonstrate some fundamental concepts that we will motivate much of our exploration.

Subsection1.1.1Some simple examples

Activity1.1.1

With a small number of unknowns, we are able to graph the sets of solutions to linear equations. Here, we will consider collections of equations having two unknowns.

  1. On the plot below, graph the lines

    \begin{equation*} \begin{split} y \amp = x+1 \\ y \amp = 2x-1 \\ \end{split} \text{.} \end{equation*}

    At what point or points \((x,y)\text{,}\) do the lines intersect? How many points \((x,y)\) satisfy both equations?

    <<SVG image is unavailable, or your browser cannot render it>>

  2. On the plot below, graph the lines

    \begin{equation*} \begin{split} y \amp = x+1 \\ y \amp = x-1. \\ \end{split} \end{equation*}

    At what point or points \((x,y)\text{,}\) do the lines intersect? How many points \((x,y)\) satisfy both equations?

    <<SVG image is unavailable, or your browser cannot render it>>

  3. On the plot below, graph the line

    \begin{equation*} y = x+1 \text{.} \end{equation*}

    How many points \((x,y)\) satisfy this equation?

    <<SVG image is unavailable, or your browser cannot render it>>

  4. On the plot below, graph the lines

    \begin{equation*} \begin{split} y \amp = x+1 \\ y \amp = 2x-1 \\ y \amp = -x. \\ \end{split} \end{equation*}

    At what point or points \((x,y)\text{,}\) do the lines intersect? How many points \((x,y)\) satisfy all three equations?

    <<SVG image is unavailable, or your browser cannot render it>>

The examples in this introductory activity demonstrate three possible outcomes, which are represented in the three figures below.

<<SVG image is unavailable, or your browser cannot render it>>

Figure1.1.1Three possibilities for collections of linear equations in two unknowns.

In this example, we see that

  • With a single equation, there are infinitely many points \((x,y)\) satisfying that equation.
  • Adding a second equation adds another condition we place on the points \((x,y)\) resulting in a single point that satisfies both equations.
  • Adding a third equation adds a third condition on the points \((x,y)\text{,}\) and it is no longer possible to satisfy all three conditions.

Generally speaking, a single equation will have many solutions, in fact, infinitely many. As we add equations, we add conditions which lead to, in a sense we will make precise later, a smaller number of solutions. Eventually, we have too many equations and find that no points satisfy all of them at the same time.

This example illustrates a general principle to which we will frequently return.

Solutions of linear equations

Given a collection of linear equations, there are either:

  • infinitely many solutions,

  • exactly one solution, or

  • no solutions.

Notice that we can see a bit more. In Figure 1, we are looking at equations in two unknowns. Here we see that

  • One equation gave us infinitely solutions.

  • Two equations gave us exactly one solution.

  • Three equations gave us no solutions.

It seems reasonable to wonder if the number of solutions depends on whether the number of equations is less than, equal to, or greater than the number of unknowns. Of course, this cannot always be the case; remember that one of our examples consisted of two equations that were graphically represented by parallel lines and that therefore had no solutions. Still, it seems safe to think that the more equations we have, the smaller the set of solutions will be.

<<SVG image is unavailable, or your browser cannot render it>>

Let's also consider some examples of equations having three unknowns, which we call \(x\text{,}\) \(y\text{,}\) and \(z\text{.}\) Just as solutions to linear equations in two unknowns formed straight lines, solutions to linear equations in three unknowns form planes.

When we consider an equation in three unknowns graphically, we need to add a third coordinate axis, as shown in Figure 2.

<<SVG image is unavailable, or your browser cannot render it>>

<<SVG image is unavailable, or your browser cannot render it>>

Figure1.1.2Coordinate systems in two and three dimensions.

As shown in Figure 3, a linear equation in two unknowns, such as \(y=0\text{,}\) is a line while a linear equation in three unknowns, such as \(z=0\text{,}\) is a plane.

<<SVG image is unavailable, or your browser cannot render it>>

<<SVG image is unavailable, or your browser cannot render it>>

Figure1.1.3The solutions to the equation \(y=0\) in two dimensions and \(z=0\) in three.

In three unknowns, the set of solutions to one linear equation forms a plane. The set of solutions to a pair of linear equations is seen graphically as the intersection of the two planes. As in Figure 4, we typically expect this intersection to be a line.

<<SVG image is unavailable, or your browser cannot render it>>

<<SVG image is unavailable, or your browser cannot render it>>

Figure1.1.4A single plane and the intersection of two planes.

When we add a third equation, we are looking for the intersection of three planes, which we expect to form a point, as in the left of Figure 5. However, in certain special cases, it may happen that there are no solutions, as seen on the right.

<<SVG image is unavailable, or your browser cannot render it>>

<<SVG image is unavailable, or your browser cannot render it>>

Figure1.1.5Two examples showing the intersections of three planes.
Activity1.1.2

This activity begins with equations having three unknowns. In this case, we know that the solutions of a single equation form a plane. If it helps with visualization, consider using \(3\times5\) inch index cards to represent planes.

  1. Is it possible that there are no solutions to two linear equations in three unknowns? Either sketch an example or give a reason why it can't happen.

  2. Is it possible that there is exactly one solution to two linear equations in three unknowns? Either sketch an example or give a reason why it can't happen.

  3. Is it possible that the solutions to four equations in three unknowns form a line? Either sketch an example or give a reason why it can't happen.

  4. What would you usually expect for the set of solutions to four equations in three unknowns?

  5. Suppose we have 500 linear equations in 10 unknowns. What would be a reasonable guess for which of the three possibilities for the set of solutions holds?

  6. Suppose we have 10 linear equations in 500 unknowns. What would be a reasonable guess for which of the three possibilities for the set of solutions holds?

Subsection1.1.2Systems of linear equations

Now that we have seen some simple examples, let's clarify what we mean by a system of linear equations.

First, we considered a linear equation having the form

\begin{equation*} y = 2x - 1\text{.} \end{equation*}

It will be convenient for us to rewrite this so that all the unknowns are on one side of the equation:

\begin{equation*} -2x + y = -1\text{.} \end{equation*}

Thinking graphically, this gives us the flexibility to describe all lines; for instance, vertical lines, such as \(x=3\text{,}\) may be represented in this form.

Notice that each term on the left is the product of a constant and the first power of a unknown. In the future, we will want to consider equations having many more unknowns, which we will sometimes denote as \(x_1, x_2, \ldots, x_n\text{.}\) This leads to the following definition:

Definition1.1.6

A linear equation in the unknowns \(x_1,x_2,\ldots,x_n\) may be written in the form

\begin{equation*} a_1x_1 + a_2x_2 + \ldots + a_nx_n = b \text{,} \end{equation*}

where \(a_1,a_2,\ldots,a_n\) are real numbers known as coefficients.

By a system of linear equations or a linear system, we mean a collection of linear equations written in a common set of unknowns. For example,

\begin{equation*} \begin{alignedat}{4} 2x_1 \amp {} + \amp {} 1.2x_2 \amp {}-{} \amp 4x_3 \amp {}={} \amp 3.7 \\ -0.1x_1 \amp {} \amp {} \amp {} + {} \amp x_3 \amp {}={} \amp 2 \\ x_1 \amp {}+{} \amp x_2 \amp {}-{} \amp x_3 \amp {}={} \amp 1.4. \\ \end{alignedat} \end{equation*}

A solution to a linear system is simply a set of numbers \(x_1 = s_1, x_2 = s_2, \ldots, x_n=s_n\) that satisfy all the equations in the system.

For instance, we earlier considered the linear system

\begin{equation*} \begin{alignedat}{3} -x \amp {}+{} \amp y \amp {} = {} \amp 1 \\ -2x \amp {}+{} \amp y \amp {} = {} \amp -1. \\ \end{alignedat} \end{equation*}

To check that \((x,y) = (2,3)\) is a solution, we verify that the following equations are valid.

\begin{equation*} \begin{alignedat}{3} -2 \amp {}+{} \amp 3 \amp {} = {} \amp 1 \\ -2(2) \amp {}+{} \amp 3 \amp {} = {} \amp -1. \\ \end{alignedat} \end{equation*}

We call the set of all solutions the solution space of the linear system.

Activity1.1.3Linear equations and their solutions

  1. Which of the following equations are linear? Please provide a justification for your response.

    1. \begin{equation*} 2x + xy -3y^2 = 2 \text{.} \end{equation*}
    2. \begin{equation*} -2x_1 + 3x_2 +4x_3 - x_5 = 0 \text{.} \end{equation*}
    3. \begin{equation*} x = 3z - 4y \text{.} \end{equation*}
  2. Consider the system of linear equations:

    \begin{equation*} \begin{alignedat}{4} x \amp {}+{} \amp y \amp \amp \amp {}={} \amp 3 \\ \amp \amp y \amp {}-{} \amp z \amp {}={} \amp 2 \\ 2x \amp {}+{} \amp y \amp {}+{} \amp z \amp {}={} \amp 4. \\ \end{alignedat} \end{equation*}
    1. Is \((x,y,z) = (1,2,0)\) a solution?

    2. Is \((x,y,z) = (-2,1,0)\) a solution?

    3. Is \((x,y,z) = (0,-3,1)\) a solution?

    4. Can you find a solution in which \(y = 0\text{?}\)

    5. Do you think there are other solutions? Please explain your response.

Subsection1.1.3Summary

The point of this section is to build some intuition about the behavior of solutions to linear systems through consideration of some simple examples. We will develop a deeper and more precise understanding of these phenomena in our future explorations.

  • A linear equation is one that may be written in the form

    \begin{equation*} a_1x_1 + a_2x_2 + \ldots + a_nx_n = b \text{.} \end{equation*}
  • A linear system is a collection of linear equations and a solution is a set of values assigned to each of the unknowns that make each equation true.

  • We came to expect that a linear system has either infinitely many solutions, exactly one solution, or no solutions.

  • When we add more equations to a system, the solution space usually seems to become smaller.