Dual space/Linear forms/Linearly independent/Orthogonal space/Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space of dimension ${}n$, let ${}f_{1},\ldots ,f_{r}\in {V}^{*}$ be linear forms, and set

${}F=\langle f_{1},\ldots ,f_{r}\rangle \subseteq {V}^{*}\,.$

Show that these linear forms are linearly independent if and only if

${}\dim _{K}{\left(F^{\perp }\right)}=n-r\,$

holds.

Create a solution