How is $v^T(Ax-b) = -b^Tv+(A^Tv)^Tx$? This might seem a silly question but I am thoroughly confused. 
Given 
$c^Tx+v^T(Ax-b) = -b^Tv+(c+A^Tv)^Tx$
I get that it can be written as the second term in the RHS, but what about the first one in the RHS. Shouldn't it be $ -v^Tb $

If $b,v$ are column vectors, $b^\top v$ is their dot product and so is $v^\top b$ as dot product is commutative. 
$$
b\cdot v=b^\top v=v^\top b=v\cdot b
$$
