Notice that we have placed the coefficients of each variable into a 2 x 1 matrix. The constant terms also appear in a 2 x 1 matrix. If we think of the matrices as vectors, we can restate the solution of the system of equations in another way:

“How can we combine the vectors of the left hand side of the equation, in just the right amounts, to get the vector on the right hand side of the equation? This right amount is called the linear combination that solves the original system of equations. The two vectors on the left hand side can be thought of as the ingredients of this particular problem. The solution is the mix of these ingredients that solves the problem.

If we take 1 of the first vector (i.e.: x = 1) and 2 of the second vector (y = 2), we get this:

In other words, the solution is or . The same solution we obtained using the Gauss Method!

We did this approach graphically on the board in class.

(1)

(2)

Before we go on, let’s solve this system of equations in two different ways that we already know about.

First Method. If we had seen these two equations in middle school, we would have said “Those are the equations of two lines. Two (non-parallel) lines intersect in a point.” And you would be right. We can indeed graph these two lines. One goes through the origin, while the other does not. Using the graphical method, we can identify the point of intersection as the point . In other words, the point is the one and only point that solves both of the equations we started with.

Second Method. (You may know this as Gaussian Elimination; if not, don’t worry ). Note that by solving this system of equations, we are finding the intersection point of two lines, just like before.

If we multiply equation (2) by 2, we can then add equations (1) and (2) together in order to eliminate the term. We are left with

which we can solve by inspection as .

Knowing that , we can substitute for in either of the equations we started with:

, or , giving us the full solution, or point of intersection as .

We are now ready to start thinking about this same problem viewed through the lens of linear algebra.

to take the time to examine this topic is a careful and systematic way.

Mathematics is the language we use to describe our observations in nature. Increasingly, linear algebra plays a fundamental role in current developments in science. Disciplines from economics and finance, straight through to biology and physics rely on linear algebra as a fundamental tool. Our explorations in this class will equally benefit students anticipating future study in these areas, as well as those interested in math as a discipline. While we will spend most time on examples and problems, we will also develop the underlying theory from time to time as well.

Science today is becoming increasingly quantitative. Scientists are being confronted more and more with large amounts of numerical data, measurements of one form or another gathered from their laboratories, field experiments (like space probes), and surveys. But merely collecting and recording data achieves nothing–the collected data must be analyzed and interpreted. Mathematics is the chief tool for this analysis and interpretation, and one of the most useful branches of mathematics for this purpose is matrix algebra.

The central element of this course–solving systems of linear equations–is pretty approachable. We are familiar with this concept reaching all the way back to middle school. Finding the intersection of two lines can be framed as a good introduction to linear algebra. We’ll take a look at this simple problem in the next post.