Dual vs reciprocal basis?

Question

I am confused as to what the differences between the two are. I have just learnt reciprocal vectors, and learnt that they are defined as $u_i v_j = \delta_{ij}$. However, I read that the dual basis is defined as $u^i v_j = \delta^i_j$. What is the difference between the two? What does the different placement of indices imply? Thank you!

Answer 1

I think the first notion is about an inner product, while the second one is the definition of dual basis of a vector space. The first is a product between elements of the same vector space, the latter is a definition of the element of the dual space: call $V$ your vector space on the field $\Bbb K$. Then by definition $V^*:=\operatorname{Hom}_{\Bbb K}(V,\Bbb K)$, and (suppose $V$ finite dimensional), given $v_1,\dots,v_n$ a basis of $V$, a basis of this space is given by the elements $v^1,\dots,v^n$ is defined by the relations $$ v^i(v_j)=\delta_{ij}. $$

The two basis are clearly connected, this is why we use the same letters; but, $v_1$ and $v^1$ are different objects, the first is a vector, the second is a function (called form), thus in order to distinguish them, we put the indexes, down for the vectors, and upper for the forms.

If it is the first time you approach this argument don't worry.