One of the fascinating things about Linear Algebra is that the prerequisites of this course are nearly nothing. When I first learn linear algebra, I was amazed by the fact that you can do so many things with just some simple integer number operation. But to catch the essence of linear algebra, we need to review some basic algebra that everybody does in grade 8: The linear equation.
We all learned the linear equations, the simple algebraic expression that somehow describes the shape of a line in the Cartesian plane. The algebra expression:
We call this slope-intercept form. We can also rearrange the equation and write down the same thing with an expression that looks like:
both forms are telling the same relation between variable \(x\) and variable \(y\) and if we draw all the pairs of \(x\) and \(y\) as a coordinate in the Cartesian plane, we will get a 2d line. all the \(a\),\(b\),\(c\),\(m\),\(b\) are just some numbers. for example:
Now let’s do some algebra. Say we have two lines, let’s call them line A and line B. the top equation is the equation for line A and line B is:
Without graphing them out, how would you find the intercept of two lines?
The question itself is fairly easy, we can use substitution or elimination to find the specific \(x\) and \(y\) value that both satisfy the equation for line A and line B, and that is the coordinate of the intercept.
Let’s start with substitution, we first take the second equation and move \(2x\) to the other side:
then we can substitute \(y\) into our first equation:
Then we grab the result and replace the \(x\) to 1 in either the first or the second equation, we can solve the \(y\):
Now let’s do the same question but with elimination. The idea is this: the goal is to solve \(x\) and \(y\), but what makes this hard to achieve is the fact that we got two variables in one equation. So, if I can get rid of one variable, we are in a good shape. Can we do that? Sure! Take equation of line B and multiply everything by a factor of 2:
Then take the equation for line A and subtract the new equation we got:
Do a little algebra we can see all the \(y\) canceled and we left with:
Repeat what we did in substitution we will end up with the same result.
2. Three-dimensional plane
Now we moved on to Three dimensions. Don’t worry if you have never studied the math in 3d space before, it is not that different from 2d. One major difference is that two variable is not enough for us to describe the location of points in 3d space. Now we need three, and let’s call the extra one \(z\). Imagine the 2d Cartesian coordinate is a blanket and you place the blanket on the floor, now \(x\) and \(y\) describe the position on the blanket and the new \(z\) coordinate describes how high the point is.
And Ta-da! We can find the coordinate of any point in three-dimensional plane! Now we know how to describe a point, some of you might ask, is the preview line equation still working in 3d? Sadly, no. The equation:
Only gives you the relation between \(x\) and \(y\), and there is no restriction on \(z\) axis, which means that this equation describes not only the line on the blanket(x-y plane), but also many different lines on top of it or below it. So what it is? The combination of all the lines is a plane. To be more specific, a 2d plane in 3d space. In general, an equation looks like:
describes a plane in 3d space.
But how do we describe a line in 3d space? You can use some math trick, say the intercepts of two planes. Or you can use what we called vector, which happens to be our first new concept.