More:
- discuss adjoints \& transposes:
  the induced action on the dual
  (properly, the map $\End V \to \End V^*$)
  Concretely, you can think of it as:
  a matrix $M$ can multiply -vectors- on the left,
  or -covectors- on the right;
  if you write the covector vertically as a vector,
  this is just the action of $M^t$ (-transpose-)
  ...and writing $w^*Av$ as $[.. ..][.. ..\\.. ..][..\\..]$
  (physicists write $\left<w|A|v\right>$;
   a mathematician might write $\left<w,v\right>_A$
   to emphasis that it's the ``pairing induced by $A$'')

- lurking in here is -tensors-:
  duals help you really understand $\Hom(V,W) = V^* \otimes W$:
  each -row- of a matrix is a covector (a coordinate);
  each -column- is a vector
