MATH0200 - LINEAR ALGEBRA

February 8, 2026

INTRODUCTION & FOUNDATIONS

One thing that I think I want to emphasize with this essay is that linear algebra really is just high-school algebra. However, the difference between linear algebra and high school algebra is that linear algebra tries to conceptualize the things that you learn in high school algebra but more geometrically, in more dimensions, with more complicated techniques, etcetera. The idea, however, still remains the same core idea, that is, dealing with linear equations. In order to conceptualize things to higher geometries and in more dimensions (like we were talking about before), we need to compress and compact the notation that we’ve already been using.

One of the things that are essential in highschool algebra is the idea of a linear equation, which has two things. 1) A line with a slope, and 2) A y-intercept. A system of linear equations are sets with more than one linear equation, and it got us thinking about how the interceptions of these lines, how two or three linear equations would interact with each other. In this, we introduced the idea of elimination and substitution: we said that setting the equations equal to each other gives us the intersection of the two lines. How did we get to this conclusion? We know that in the equation y=mx+b, there is ultimately a relationship between two variables: x and y. Meaning, some number that we have is transformed by something, and the way that we can graph that transformation, or deviation from an original value y, is by plotting what the number used to be and then what the number ended up being. This ultimately forms the coordinate system.

Well, how did we get to the point that equating something to something else gives us the intersection? If we rethink of the y=x equation that we’re used to into the idea of an original value (y) and a deviated value (x), we can say that whenever we’re trying to find the , we’re focusing on if we have a certain “original value” that we want to get, and there is supposedly two transformations that give us the same original value, then we set the two transformations equal to each other. What substitution and elimination does is it generalizes the act of “setting two transformations together” into little tricks, like multiplying the equation by a scalar and everything, because that operation doesn’t change anything about the actual equation that we’re working with.

ESTABISHING A METHOD OF LINEAR ALGEBRA

We saw that whenever we wanted to do something more efficiently, we invented new techniques that worked just as good as the “technically correct and longer system” but a little faster. However, the main thing to understand with this is that we’re rearranging

dongnimmun

dongnimmun