Finite-dimensional vector space/Dual basis/Basis/Fact/Proof

Proof

Suppose that

${}\sum _{j=1}^{n}a_{j}v_{j}^{*}=0\,,$

where ${}a_{j}\in K$. If we apply this linear form to ${}v_{i}$, we get directly

${}a_{i}=0\,.$

Therefore, the ${}v_{1}^{*},\ldots ,v_{n}^{*}$ are linearly independent. Due to fact, the dual space has dimension ${}n$, thus we have a basis already.

To fact