Find the least squares solution for rank deficient system 
Find the least squares solution to the system
$$x - y = 4$$
$$x - y = 6$$

Normally if I knew what the matrix $A$ was and what $b$ was I could just do $(A^TA)^{-1} A^Tb$, but in this case I'm not sure how to set up my matrices. How can I find the least square solution to the system?

Your matrix is just the coefficients of your system of equations.  In this case 
$$ x-y = 4  $$
$$ x-y = 6  $$
leads to 
$$
\begin{bmatrix}
1 & -1 \\ 1 & -1 
\end{bmatrix}
\begin{bmatrix}
x \\ y
\end{bmatrix} 
=
\begin{bmatrix}
4 \\ 6
\end{bmatrix}
$$
but you should see that there is no solution to this since you can't have $x-y$ be both $4$ and $6$...
