-1
$\begingroup$

In the finite dimensional vector space $V$, suppose $\{f_1,f_2,\cdots,f_m\}$ are the dual basis, how can find the basis $\{e_1,e_2,\cdots,e_m\}$ s.t. $f_i(e_j)=\delta_{ij}$

$\endgroup$

closed as unclear what you're asking by Micah, Mark Viola, qwr, Semiclassical, Matt Samuel Jul 8 '15 at 1:32

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Can you elaborate on what you mean by "find?" Are you asking whether they exist? Or, how are you writing down the $f_i$? $\endgroup$ – user148177 Jul 6 '15 at 6:55
  • $\begingroup$ Same question here and here $\endgroup$ – Noix07 Feb 28 '16 at 20:00
1
$\begingroup$

If you have the finite dimensional vector space V, and consider the dual basis is $B^{*}=${$f_1,...,f_m$} then to find the basis $B=${$\vec{e_1},...,\vec{e_m}$} you only have to apply that $f_i(e_j)=\delta_{ij}$

example:

to fing $\vec{e_1}$ then you have a linear system

$f_1(\vec{e_1})=1$ and $f_1(\vec{e_2})=0$ ... $f_1(e_m)=0$ there you can find the componets of the vector $\vec{e_1}$, so you know who is the vector $\vec{e_1}$

to find $\vec{e_m}$ repeat

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.