Geometric interpretation of linear systems

In the previous chapter we studied equations of lines, planes, etc. As you might already realize, it was just a lead-in for the geometric interpretation of linear systems. Let's start!

Linear systems

Any linear system consists of a set of equations. Since the system is linear, then obviously every equation in it is also linear, which means that every equation defines a certain geometric entity, as we have found out in the previous topic. If a linear equation has $2$ unknowns then it defines a line, and if it has $3$ unknowns, then it defines a three-dimensional plane. It's easy to imagine those two entities, but there are also linear equations with higher variable count, which define different hyperplanes. The information given further will work for any multidimensional space, but, since it's much easier to imagine $2D$ and $3D$ spaces both in our heads and on paper, from this point on we will have them in mind.

Two-dimensional space

In the two-dimensional space we can define a line. The equation of a line has the following form: $ax+by+c=0$. In order to make it look more similar to equations in linear systems, let's move the intercept to the right side with the minus sign, and leave the unknowns in the left side $ax+by=-c$. This is exactly how an equation in a linear system looks like.
The solution for a linear system with two unknowns will be the intersection of all the lines. It's clear that lines can be different and so can solutions: it could be a line, a dot, or nothing. If there isn't a place on the graph where all lines intersect, then the equation system doesn't have a solution. Now let's examine it in detail on some examples.

Consider the following system:

$$\begin{cases} 4x-3y=-2\\ -8x+3y=-2 \end{cases}$$

Let's draw the two lines on the plane and see what happens:

We can see that the lines (the first one is red, the second one is blue) intersect at $(1,2)$. That will be the solution for this equation system.

Now let's take a look at this example:

$$\begin{cases} x-y=-2\\ 3x-3y=-6 \end{cases}$$

After drawing the graph we will notice that the lines are identical:

Which is true, since the equations in the system only differ by a coefficient $3$, that's why they define the same line. The solution for this system is the whole line $x-y=-2$, so there is an infinite number of solutions. This time consider the opposite situation. Imagine, we are given this linear system:

$$\begin{cases} x-y=-2\\ x-y=0 \end{cases}$$

Draw it on a graph:

The graph shows that the lines are parallel and thus don't intersect. Therefore, such a system doesn't have any solutions.

It's important to remember that if the linear system has, for example, three equations, then the solution will be the place where ALL three lines intersect.

Three-dimensional space

In three-dimensional space there are planes, lines and dots. Therefore, a solution of a linear system with three unknowns could be a plane, a line, a dot or nothing. The equation for a three-dimensional plane looks like this: $ax+by+cz=-d$

If two planes are not parallel, then the solution will be a line, as it is illustrated below:

If two planes are identical, then the solution will be the whole plane (similarly to the two-dimensional space). It's also not hard to understand that if two planes are parallel and aren't identical, then the system will not have any solutions, since the planes will never intersect.

Let's also consider one more case. Say, the system has three equations. The solution for the system would be a space that belongs to all three planes at the same time. So if the first two planes intersect on a line, then the solution will be the intersection of the said line with the third plane, which, based on the coefficients, will be either a dot or a line.

Hyperspace

Imagine, that we have $n$ unknowns in the linear system. It means that every equation defines a $n$-dimensional plane. It's hard to draw an image of this in your mind, but it's actually rather simple to understand. After intersecting of two $n$-dimensional planes we will get a $(n-1)$-dimensional plane. So everything goes the same way as with two- or three-dimensional planes.

Conclusion

In this chapter we have found out about the geometric interpretation of linear systems. Unfortunately, systems with a large number of unknowns are difficult to imagine or illustrate, but it's easier for two- and three-dimensional planes.