How to expand in a non-orthogonal basis in an intuitive way? I have a basis that consists of four non-orthogonal vectors $\{|u_i\rangle\}, 1 \le i \le 4$. Can the formula for an orthonormal expansion be modified so that it holds true for any given basis?
$$|v>=\sum_i |u_i\rangle\langle u_i|v\rangle$$
I could always write $|v\rangle$ as a linear conbination of $\{u_i\}$ and solve the equation system, but I would like to approach the problem from a different perspective. I am not interested in finding an orthonormal basis with the Gram–Schmidt process.
 A: Let me write vectors without using bra-ket notation because I'm not familliar with it. Also let there be just 2 basis vectors, for ease of typing. Let $\langle a,b\rangle$ denote the inner product of $a,b$.
You are asking for the correct coefficients $a_1,a_2$ for the expansion
$$ v = a_1 u_1 + a_2 u_2$$ 
without using explicitly a change of basis from an original known basis, expressed solely in terms of $\langle v,u_i\rangle$. By taking inner products with the $u_i$, this is the same problem as trying to solve
$$ \mathbf v = M\mathbf a $$
where:
$$ \mathbf v = \binom{\langle u_1,v\rangle}{\langle u_2,v\rangle},\quad \mathbf a = \binom{a_1}{a_2},\quad M =\begin{bmatrix} \langle u_1, u_1\rangle &  \langle u_1,u_2\rangle \\
\langle u_2, u_1\rangle &  \langle u_2,u_2\rangle  \end{bmatrix}.$$
So the coefficients are given by $M^{-1}\mathbf v$. With just 2 vectors, this is easy to write down explicitly,
$$ \mathbf a = \frac1{\langle u_1, u_1\rangle \langle u_2, u_2\rangle - \langle u_1, u_2\rangle\langle u_2, u_1\rangle } 
\binom
{\phantom{+}\langle u_2, u_2\rangle \langle u_1, v\rangle 
- \langle u_1, u_2\rangle \langle u_2, v\rangle}
{-\langle u_2, u_1\rangle \langle  u_1,v\rangle 
+ \langle u_1, u_1\rangle \langle u_2,v\rangle}$$
As requested, this is not written in terms of the canonical basis. It does involve a matrix in the derivation but once derived, the formula is plug-and-play. It is clear how the same idea works for any number of basis vectors.
Remark: this matrix $M$ is known as the Gramian matrix of the vectors $u_1,\dots,u_n$.
A: In co-ordinate-free terms, as long as you define your bras $\langle u_i|$ so that $\langle u_i|u_j \rangle = \delta_{ij}$ then if
$| v \rangle = \sum_j \lambda_j |u_j\rangle $
you have
$\langle u_i| v \rangle = \sum_j \lambda_j \langle u_i|u_j\rangle = \sum_j \lambda_j \delta_{ij} = \lambda_i$
so
$|v \rangle = \sum_i |u_i\rangle \langle u_i| v \rangle$
In co-ordinate terms, if you are expressing your kets $|u_i\rangle$ as co-ordinates $|u_i\rangle = \{u_{ij}|1\le j\le 4\}$ realtive to some basis then expressing the bras $\langle u_i|$ in the same basis is equivalent to finding the inverse of the $4 \times 4$ matrix formed by the $\{u_{ij}\}$.
