# The Unapologetic Mathematician

## The Matrix of a Linear System

As I wait for the iTunes store to be less busy so it can reauthorize my iPhone to work with the updated firmware, we can finally get back on track.

Let’s consider a system of linear equations. We’ll use the $m$ variables $x^1$, $x^2$, and so on up to $x^m$; and we’ll let there be $n$ equations. Let’s write these out:

$a_1^1x^1 + a_2^1x^2 + ... + a_m^1x^m = y^1$
$a_1^2x^1 + a_2^2x^2 + ... + a_m^2x^m = y^2$

$a_1^nx^1 + a_2^nx^2 + ... + a_m^nx^m = y^n$

Here the constant $a_i^j$ are the coefficient of $x^i$ in the $j$th equation, and $y^j$ is the constant term on the right hand side of the $j$th equation.

But this is all but writing out exactly our matrix notation! We can take the above system and rewrite it as

$\displaystyle\begin{pmatrix}a_1^1&a_2^1&\cdots&a_m^1\\a_1^2&a_2^2&\cdots&a_m^2\\\vdots&\vdots&\ddots&\vdots\\a_1^n&a_2^n&\cdots&a_m^n\end{pmatrix}\begin{pmatrix}x^1\\x^2\\\vdots\\x^m\end{pmatrix}=\begin{pmatrix}y^1\\y^2\\\vdots\\y^n\end{pmatrix}$

Picking values for the variables $x^i$ is the same as picking the components of a column vector $x=x^ie_i$. We can collect the right hand sides of all our equations into one column vector $y=y^jf_j$, and the coefficients give a (linear) formula for taking the values we choose for the variables and turning them into the $n$ values on the right of our equations. That is, they define a linear map $A:\mathbb{F}^m\rightarrow\mathbb{F}^n$. We can thus rewrite our system in a more abstract notation as:

$Ax=y$

Suddenly it looks a lot more like the first — and simplest — linear equation we wrote down. But now we can’t just “divide by $A$” to solve it. We need heavier tools to manage this task, or even just to show when it can be managed at all! In short: we need linear algebra.

Incidentally, now we see why we indexed the variables with superscripts: because that’s how we wrote the components of a vector, and the variables are the components of a single vector variable. And if you’re still on the fence, I’ll note that physicists use superscripts all the time to index variables (for similar purposes), and they even do it when the equations aren’t all linear. Just try it. You’ll get used to it.

July 11, 2008 - Posted by | Algebra, Linear Algebra

1. The justification that comes to my mind for using superscripts is that the function “the nth component of this vector” with respect to a given basis is a linear functional, so it belongs in the dual space. Similarly, a linear transformation is an element of $V \otimes V^{*}$ – one superscript and one subscript.