Science:Math Exam Resources/Courses/MATH221/December 2019/Question 06 (c)/Statement

Let $V$ be a subspace of $\mathbb {R} ^{7}$ and let ${\bf {x}}$ be a vector in $\mathbb {R} ^{7}.$ Suppose

$${\bf {v}}_{1}+{\bf {u}}_{1}={\bf {x}}={\bf {v}}_{2}+{\bf {u}}_{2},$$

where ${\bf {v}}_{1},{\bf {v}}_{2}$ are in $V$ and ${\bf {u}}_{1},{\bf {u}}_{2}$ are in $V^{\perp }$. Show that ${\bf {v}}_{1}={\bf {v}}_{2}.$