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

Let $V$ be a finite-dimensional $K$-vector space of dimension $n$, let $f_1,\dots ,f_r\in V^{\ast }$ be linear forms, and set

$F=⟨f_1,\dots ,f_r⟩\subseteq V^{\ast }.$

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

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

holds.

Create a solution