Change of basis and inner product in non-orthogonal basis I have a vector, originally expressed in the standard coordinates system, and want to perform a change of basis and find coordinates in another basis, this basis being non-orthogonal.

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*Let $B = \{e_1, e_2\}$ be the standard basis for $\Bbb R^2$.

*Let $B' = \{e_1', e_2'\}$ be a non-orthogonal basis for $\Bbb R^2$.

*Let $v$ be some vector in $\Bbb R^2$.

The standard inner product is
$\langle a, b \rangle = \sum_{i=0}^n a_i b_i.$
I want to define an inner product in the non-orthogonal basis $B'$ so that $\langle e_1', e_2' \rangle_{B'} = 0$ since  $\sum_{i=0}^n e_{1i}' e_{2i}' \neq 0$.
Basically, I want to use this new inner product to get the component/coordinates of the vector $v$ on the basis $B'$.


Let u,v be any lin ind vectors in the plane. Define u.u=a>0,u.v=0=v.u,v.v=b>0. Extend this linearly for all pairs of vectors. It is easily verified that this is an inner product.  
