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

The matrix $A$ and the vectors ${\mathbf{𝐯}}_1,{\mathbf{𝐯}}_2$ and ${\mathbf{𝐯}}_3$ are the same as in the previous part. Let $W$ be a 2-dimensional subspace of ${\mathbb{ℝ}}^3$ such that, whenever a vector $\mathbf{𝐱}$ is in $W$, the vector $A\mathbf{𝐱}$ is also in $W$. Suppose a vector

$$\mathbf{𝐱}=c_1{\mathbf{𝐯}}_1+c_2{\mathbf{𝐯}}_2+c_3{\mathbf{𝐯}}_3$$

is in $W$ where $c_1\ne 0$ and $c_2\ne 0$. Show that $c_3=0.$