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. $f_i(e_j)=\delta_{ij}$


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  • $\begingroup$ 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

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 $f_i(e_j)=\delta_{ij}$


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


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