Let $F$ be an isomorphism of euclidian space $E$, with orthonormal basis $\{e_1,\ldots e_n\}$. Let $F'$ be orthogonalised $F$.
Is any operator $F_t$ from linear homotopy of $F$ and $F'$ an isomorphism ($F_t=tF'+(1-t)F$, $t\in [0;1]$)?
A little explanation on $F':$ let $\{f_1,\ldots f_n\}$ be an image of $\{e_1,\ldots e_n\}$ under $F$, $f_i:=F(e_i)$, we then apply Gram–Schmidt process (without norming), and get $\{f_1',\ldots f_n'\}$. $F'$ is defined as $F'(e_i)=f_i'$