Show that, over an arbitrary field K {\displaystyle {}K} , for linearly independent vectors u = ( a b c ) {\displaystyle {}u={\begin{pmatrix}a\\b\\c\end{pmatrix}}} and v = ( d e f ) {\displaystyle {}v={\begin{pmatrix}d\\e\\f\end{pmatrix}}} , the family consisting of u , v {\displaystyle {}u,v} , and the cross product ( a b c ) × ( d e f ) {\displaystyle {}{\begin{pmatrix}a\\b\\c\end{pmatrix}}\times {\begin{pmatrix}d\\e\\f\end{pmatrix}}} does not necessarily form a basis of K 3 {\displaystyle {}K^{3}} .
