= Dual space
{title2=$V^*$}
{wiki}
The dual space of a <vector space> $V$, sometimes denoted $V^*$, is the vector space of all <linear forms> over $V$ with the obvious addition and scalar multiplication operations defined.
Since a linear form is completely determined by how it acts on a \x[basis]{magic}, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the $V$, and so they are isomorphic because <all vector spaces of the same dimension on a given field are isomorphic>, and so the dual is quite a boring concept in the context of finite dimension.
Infinite dimension seems more interesting however, see: https://en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278\#Infinite-dimensional_case
One place where duals are different from the non-duals however is when dealing with <tensors>, because they transform differently than vectors from the base space $V$.