TITLE: Understanding Dual Transformations and reasoning behind definition
QUESTION [6 upvotes]: In Linear Algebra working with Dual space and dual transformations I've come along this very basic definition of the dual transformations:

Suppose:
$T^*$ is a dual transformations from $W^*\to V^*$
   $T$ is a linear transformation from $V\to W$
   $u$ is a linear functional which belongs to $W^*$
   $v$ is a vector which belongs to $V$

   Then the following applies:
     $$ (T^*u)(v)  = u(Tv) $$ 

Why is the dual transformation defined this way? (I know this is a very problematic question, but please any intuition will be very helpful)

REPLY [8 votes]: In the special case of matrix algebra, this turns out to be fairly obvious.
In this setting, one usually writes vectors as $n \times 1$ matrices ("column vectors"), and linear functionals as $1 \times n$ matrices ("row vectors").
If we have a matrix $A$ with suitable dimensions, then "multiplication on the left" results in a linear transformation (call it $T$): i.e. $T(v) = Av$. The dual transformation is "multiplication on the right". That is, $T^*(u) = uA$. So your identity is merely
$$ (uA)v = u(Av) $$