I'm trying to teach myself general relativity, and I came across the subject of covariant and contravariant components of a vector. Suppose your basis is $e_1, e_2$, and they are separated by 45°. Let the magnitude of $e_1$ be 2, and let the magnitude of $e_2$ be 1. My question is how do you find the dual basis $e^1, e^2$?
I set up the following matrix equation:
$$e^2 =\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix} \begin{matrix}2e_1\\e_2\end{matrix}$$
But that's where I'm stuck. I know that after you rotate $e_1$ by 90° you have to rescale so that $e^2 \cdot e_2 = 1$, but I can't see how to rotate $e_1$ by 90 degrees.
I read in Wikipedia: In a finite-dimensional vector space V over a field K with a symmetric bilinear form g : V × V → K (which may be referred to as the metric tensor), there is little distinction between covariant and contravariant vectors, because the bilinear form allows covectors to be identified with vectors. That is, a vector v uniquely determines a covector a through the equation a(w)=g(v,w) for all vectors w. Conversely, each covector a determines a unique vector v by this equation. Because of this identification of vectors with covectors, one may speak of the covariant components or contravariant components of a vector, that is, they are just representations of the same vector using the reciprocal basis.
I'm trying to find the reciprocal basis for $e_1, e_2$.
