Let V {\displaystyle V} be a subspace of R 7 {\displaystyle \mathbb {R} ^{7}} and let x {\displaystyle {\bf {x}}} be a vector in R 7 . {\displaystyle \mathbb {R} ^{7}.} Suppose v 1 + u 1 = x = v 2 + u 2 , {\displaystyle {\bf {v}}_{1}+{\bf {u}}_{1}={\bf {x}}={\bf {v}}_{2}+{\bf {u}}_{2},} where v 1 , v 2 {\displaystyle {\bf {v}}_{1},{\bf {v}}_{2}} are in V {\displaystyle V} and u 1 , u 2 {\displaystyle {\bf {u}}_{1},{\bf {u}}_{2}} are in V ⊥ {\displaystyle V^{\perp }} . Show that v 1 = v 2 . {\displaystyle {\bf {v}}_{1}={\bf {v}}_{2}.}
