Let \(\displaystyle R:\bold{R}^3 \rightarrow \bold{R}^3\) by perpendicular reflection across the plane \(\displaystyle x+2y-3z=0\)
i) Find two independent vectors in \(\displaystyle \bold{R}^3\) which map onto themselves.
ii) Find a vector in \(\displaystyle \bold{R}^3\) which map onto its opposite.
In the last 5 minutes of my last class we flew through an \(\displaystyle \bold{R}^2\) version of this problem, but I do not have a grasp on it at all. Maybe if someone can explain in more detail how to acquire these two, that would be great. I know (i) is searching for a basis, but I am unsure of the method I should use to get the solution.
i) Find two independent vectors in \(\displaystyle \bold{R}^3\) which map onto themselves.
ii) Find a vector in \(\displaystyle \bold{R}^3\) which map onto its opposite.
In the last 5 minutes of my last class we flew through an \(\displaystyle \bold{R}^2\) version of this problem, but I do not have a grasp on it at all. Maybe if someone can explain in more detail how to acquire these two, that would be great. I know (i) is searching for a basis, but I am unsure of the method I should use to get the solution.