Finite-dimensional vector space/Dual basis/Definition/Explanations/Remark

Because of fact, this rule defines indeed a linear form. The linear form ${}v_{i}^{*}$ assigns to an arbitrary vector ${}v\in V$ the ${}i$-th coordinate of ${}v$ with respect to the given basis. Note that for ${}v=\sum _{j=1}^{n}s_{j}v_{j}$, we have

${}v_{i}^{*}(v)=v_{i}^{*}{\left(\sum _{j=1}^{n}s_{j}v_{j}\right)}=\sum _{j=1}^{n}s_{j}v_{i}^{*}(v_{j})=s_{i}\,.$

It is important to stress that ${}v_{i}^{*}$ does not only depend on the vector ${}v_{i}$, but on the basis. There doe not exist something like a "dual vector“ for a vector. This looks different in the situation where an inner product is given on ${}V$.