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 $(e_1^*, e_2^*, ...., e_n^*)$?

Let $V$ be a vector space over a field $k$ with basis $e_1, ...  e_n$.  The dual $V^{\ast}$ is defined to be the vector space consisting of all linear transformations from $V$ to $k$.  The definitions of addition and scalar multiplication in $V^{\ast}$ are obvious, as is the fact that $V^{\ast}$ is actually a vector space.
Every element in $V$ can be uniquely written as a sum $c_1e_1 + \cdots + c_ne_n$, where $c_i$ are in $k$.  For each $j$, define a function $e_j^{\ast}: V \rightarrow k$ by the formula $$e_j^{\ast}(c_1e_1 + \cdots + e_nv_n) = c_j$$  You can check that for any $x, y \in V$, and any $c \in k$, that $e_j^{\ast}(x+y) = e_j^{\ast}(x) + e_j^{\ast}(y)$, and $e_j^{\ast}(cx) = c e_j^{\ast}(x)$.  In other words, $e_j^{\ast}$ is a linear transformation from $V$ to $k$, so $e_j^{\ast} \in V^{\ast}$.
The basic result is that the elements $e_i^{\ast}$ form a basis for the vector space $V^{\ast}$.  Is this what you're having trouble proving?
