>
  </div>As you completed the problems above, did you notice any
  connections between the size of the matrix and the size of the
  input and output vectors? Make sure you ask in class about this.
  We'll make a connection. We've now seen that the derivative of
  $z=f(x,y)$ is a matrix $Df(x,y) = \begin{bmatrix}f_x &amp;
  f_y\end{bmatrix}$. This means that $Df$ is itself a function from
  $\mathbb{R}^2$ to $\mathbb{R}^2$ that has inputs $x$ and $y$ and
  outputs $f_x$ and $f_y$. Therefore we can draw $Df$