In page $80$ of this book, Exercise 4.11 (Rank factorization theorem) proves that every $m\times n$ matrix $A$ of rank $r$ can be written as the sum of $r$ matrices of rank $1$. In the proof, the author eventually states that $$ Ae_i=a_i=B(C'e_i)=(BC')e_i $$ for all $i$.
My question is: what does $e_i$ mean in this proof?
$e_i$ is the $i$th member of the standard basis, that is $e_1=(1,0,\ldots,0)^T$ and so on until $e_n=(0,\ldots,0,1)^T$. Or, in analogy to the notation $a_i$ they use for the $i$th colun of matrix $A$, you may view $e_i$ as the $i$th column of the identity matrix.