deletions | additions       diff --git a/Complex.tex b/Complex.tex  index d261545..aededb1 100644  --- a/Complex.tex  +++ b/Complex.tex 
...
x=\Upsilon\# b  \end{equation}  However it is not yet clear how to define the $\#$ operation to a vector. This can be explored, we must find the general expression for a matrix to matrix $\#$ operation. In the mean time we can set up a scenario we know the solution to, attempting to reverse engineer the result! \begin{equation}  \begin{bmatrix}n&0\\0&n\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}nx_1\\nx_2\end{bmatrix}  \end{equation}  We know the $\#$ inverse of the identity matrix,  \begin{equation}  \frac{1}{n}\begin{bmatrix}1&0\\1&0\end{bmatrix}\#\begin{bmatrix}n&0\\0&n\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\frac{1}{n}\begin{bmatrix}1&0\\1&0\end{bmatrix}\#\begin{bmatrix}nx_1\\nx_2\end{bmatrix}  \end{equation}  The factors of $n$ will cancel, however this reveals to us that the left identity is still an identity for a vector. This is fairly obvious, but now known for sure. A less easy example would be   \begin{equation}  \begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}ax_1+bx_2\\cx_1+dx_2\end{bmatrix}  \end{equation}  We can use our $\#$ inverse already found to give \begin{equation}  \bigg(\frac{1}{ac+bd}\begin{bmatrix}c & -d \\ a & -b \end{bmatrix}\#\begin{bmatrix}a&b\\c&d\end{bmatrix}\bigg)\begin{bmatrix}x_1\\x_2\end{bmatrix}=\frac{1}{ac+bd}\begin{bmatrix}c & -d \\ a & -b \end{bmatrix}\#\begin{bmatrix}ax_1+bx_2\\cx_1+dx_2\end{bmatrix}\\  \begin{bmatrix}x_1\\x_2\end{bmatrix}=\frac{1}{ac+bd}\begin{bmatrix}c & -d \\ a & -b \end{bmatrix}\#\begin{bmatrix}ax_1+bx_2\\cx_1+dx_2\end{bmatrix}  \end{equation}