Having to do copious calculations by hand when preparing for an exam, I came to realize that there was an alternative way of interpreting a matrix multiplication. This new insight would allow me to instantly guess the following product without ever doing any numerical multiplication:
\[\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & -8 & 0
\end{bmatrix}
\begin{bmatrix}
0 & 0 & 0 \\
0& 1 & 0 \\
1& 0 & 0
\end{bmatrix}
=
\begin{bmatrix}
3 & 2 & 0 \\
6 & 5 & 0 \\
0 & -8 & 0
\end{bmatrix}\]

Was there a way to have known that the first column of the product would be the third column of the first matrix?

Continue reading “Matrix Multiplication, the human way!” »

The real numbers have an ordering on them–given two numbers and , we can tell whether  or . So as math people, we like to generalize this to other sets–when can we say that a general set is ordered? In this post, we’re going to explain the explicit of idea of what it means for a field to be ordered, and then show that the complex numbers  can’t be ordered–no matter what ordering you put on it. (If you don’t know what a field is, just think of the real numbers or the complex numbers .)

Continue reading “Ordered Fields and When You Can’t Order Them” »