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 $(e_1^*, e_2^*, …., e_n^*)$?
[Math] Finding dual basis to some given basis
linear algebra
Best Answer
Put the vectors $e_i$ in the columns of some matrix $A$. If you have a matrix $B$ having as columns the vectors $e_i^*$, then $B^*A=I$, where $B^*$ is the transpose of $B$. Hence, $B=(A^{-1})^*$.
So, the elements of the dual basis are the columns of the matrix $(A^{-1})^*$.