Science:Math Exam Resources/Courses/MATH221/December 2019/Question 10 (b)/Statement

The matrix $A$ and the vectors ${\bf {v}}_{1},\;{\bf {v}}_{2}$ and ${\bf {v}}_{3}$ are the same as in the previous part. Let $W$ be a 2-dimensional subspace of $\mathbb {R} ^{3}$ such that, whenever a vector ${\bf {x}}$ is in $W$, the vector $A{\bf {x}}$ is also in $W$. Suppose a vector

$${\bf {x}}=c_{1}{\bf {v}}_{1}+c_{2}{\bf {v}}_{2}+c_{3}{\bf {v}}_{3}$$

is in $W$ where $c_{1}\neq 0$ and $c_{2}\neq 0$. Show that $c_{3}=0.$