# Relationship of matrices with geometry

I have recently started learning linear algebra. Although I have studied in high school but that was just solving equations without knowing the purpose.

Coming straight to my question, for the following system of linear equations.

x1 + 2x2 = 5
2x1 + 3x2 = 8


After finding the value of x1 and x2 they are plotted on the graph. Also, for the system

x1 + x2 = 2
x1 + x2 = 1


Author has written it has no solution as lines on the graph are parallel.

Why do we get parallel lines if system has no solution? Why after solving linear systems they are plotted on graphs? If we have solution plotted on 2D plan, can we write linear system from that?

This is my first attempt in learning mathematics. If I use wrong terms please correct me.

• Each equation in the system defines set of points we call line (hence linear). When system has one solution, it means that a point belongs to both sets of points (lines intersect). When system has infinitely many solutions, both sets are the same so the lines coincide. And the third case is when lines are parallel - no solution. – Vasya May 21 at 22:45
• Matrices are not a key self-sufficient object in linear algebra. To understand the geometry, you want to study linear transformations, and first vector spaces, bases, and other concepts. Use Halmos's Finite-dimensional vector spaces for this: it gives the clean geometry, and only then introduces coordinates. (Introducing coordinates too soon causes the kind of confusion you are experiencing.) – avs May 21 at 22:56

With systems of linear equations, the system can be described as ONE of the following:

1-Has one solution as in the first case.

2-Has no solution as in the second case.

3-The system is inconsistent. For example: $$x+y=2$$ $$x+y=3$$

4-Has Infinite number of solutions. This is the case when we have more equations than we have variables. In the case of 2 variables, you can have the following "system": $$x+2y=5$$ ONLY. There are many (infinite number in fact) of (x,y) pairs that satisfy this last equation by itself. Such system has infinite number of solutions.

"Why do we get parallel lines if system has no solution?"


The word solution means the point (x,y) of intersection. So, if the lines are parallel there is no intersection, and hence no solution.

"Why after solving linear systems they are plotted on graphs?"


This is not a must unless someone explicitly asks you to. Visualizing things is easy these days and provide better understanding with little effort. However, it is optional unless someone asks for it. It is not always possible. If you have a system of 100 equations, well this is not possible to plot.

"If we have solution plotted on 2D plan, can we write linear system from that?"


Yes we can. For each line, we take any 2 points, find the equation of the line passing throug the 2 points and do the same for the other line. This way you get your 2-D system built from a plot/graph.

By identifying $$(x_1,x_2)=(x,y)$$ we can consider a system of linear equations such as $$x_1+2x_2=5$$ $$2x_1+3x_2=8$$ as the following $$x+2y=5$$ $$2x+3y=8$$ system of lines in $$\mathbb{R}^2$$. A single solution to the above system consists of an ordered pair $$(x,y)$$. In other words, the solution (if it exists) of the above system geometrically represents where the two lines intersect in $$\mathbb{R}^2$$ as an ordered pair.

The system $$x_1+x_2=2$$ $$x_1+x_2=1$$ corresponds to the system $$x+y=2\iff y=2-x$$ $$x+y=1\iff y=1-x$$

We see that the equations on the right represent two lines in $$y=mx+b$$ form with the same slope of $$-1$$. The lines differ only in their $$y$$-intercepts. Because the lines' slopes are the same, they are parallel lines. But since these lines are not the same line, but are just parallel, this implies the lines will never intersect (Euclid) and hence there will be no solution.

• How intersection point on 2D plane ends up being a solution? – mallaudin May 21 at 23:01
• What do you mean by 'how' and 'ends up'? – coreyman317 May 21 at 23:03
• What if I have three unknowns x1, x2 and x3. How do you map them to x,y? – mallaudin May 22 at 6:19
• If there are three variables $(x_1,x_2,x_3)$ then you can 'think' of them as representing the point $(x,y,z)$ in $\mathbb{R}^3$. Then the geometric picture is about lines and planes. – coreyman317 May 22 at 6:22