How do I convert a summation equation to a vector equation (backpropagation)? $$a_j^l=\sigma(\sum_{k} {w_j}_k^l {a}_k^{l-1}+b_j^l)$$
$$a^l=\sigma( w^l {a}^{l-1}+b^l)$$
In a resource I have been reading, the above equations describe the activation of a neurone. They have the same function (I agree) but the first is a summation representation, whereas the second is this equation but in a vector form. My questions are, What did they do to remove the subscripts, and what is the process for converting summation equations to vector equations?
$a_j^l$ is the $j$-th component of $a^l$. Hence, the first equation is $1$ dimensional, the second equation is a vector equation. 

Similarly for notaiton $b_j^l$ and $w_{jk}$.

$\sigma$ means sigmoid function. If the input is a vector, then it evaluated the vector componentwise.

To answer your main quesiton, recall that from definition of  matrix vector product, where $A$ is a matrix and $x$ is a vector, $Ax$ produces a vector $y$ where the $j$-th component is $y_j= \sum_k A_{jk}x_k$.  

That is element of $$y=Ax$$

can be written as $$y_j = \sum_k A_{jk}x_k$$

elementwise.

Hence that is how we can "remove" the summation sign.