Finding dual basis to some given basis Can someone briefly explain me how to find dual basis to some given basis?
Let's say we've got $n$-dimensional linear space with given basis $(e_1,e_2,...,e_n)$. How to find $(e_1^*, e_2^*, ...., e_n^*)$?

Okej, so let's suppose we've got $([1,1,1],[1,1,-1],[1,-1,-1])$, some basis of $\mathbb{R}^3$. That's how I can generate dual basis:
Let
\begin{bmatrix}
1 &1  &1 \\ 
1 &1  &-1 \\ 
1 &-1  &-1 
\end{bmatrix}
be the $A$ matrix. I have to find $A^{-1}$, which is
\begin{bmatrix}
0,5 &0  &0,5 \\ 
0 &0,5  &-0,5 \\ 
0,5 &-0,5  &0 
\end{bmatrix}
And that's all. Dual basis is $((\frac{1}{2}x^2+\frac{1}{2}), (\frac{1}{2}x -\frac{1}{2}), (\frac{1}{2}x^2-\frac{1}{2}x))$
Right? And I cannot avoid calculation $A^{-1}$, right?
