Let K {\displaystyle {}K} be a field, and let V {\displaystyle {}V} be a K {\displaystyle {}K} -vector space of finite dimension. Let U ⊆ V {\displaystyle {}U\subseteq V} denote a linear subspace with dim K ( U ) = dim K ( V ) {\displaystyle {}\dim _{K}{\left(U\right)}=\dim _{K}{\left(V\right)}} . Show that U = V {\displaystyle {}U=V} holds.
