Let U 1 = ⟨ ( 4 − 8 7 ) ⟩ {\displaystyle {}U_{1}=\langle {\begin{pmatrix}4\\-8\\7\end{pmatrix}}\rangle } and U 2 = ⟨ ( 4 − 8 7 ) , ( 3 − 5 2 ) ⟩ {\displaystyle {}U_{2}=\langle {\begin{pmatrix}4\\-8\\7\end{pmatrix}},\,{\begin{pmatrix}3\\-5\\2\end{pmatrix}}\rangle } be linear subspaces in R 3 {\displaystyle {}\mathbb {R} ^{3}} . Find an isometry φ : R 3 → R 3 {\displaystyle {}\varphi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} fulfilling U 1 = ⟨ φ ( e 1 ) ⟩ {\displaystyle {}U_{1}=\langle \varphi (e_{1})\rangle } and U 2 = ⟨ φ ( e 1 ) , φ ( e 2 ) ⟩ {\displaystyle {}U_{2}=\langle \varphi (e_{1}),\,\varphi (e_{2})\rangle } .
