I am practicing some linear algebra question to prepare for my test. I have come across one question that has given me much trouble. It states:
If $\lVert u\rVert = 1$, then $Q = I - 2uu^T$ is a reflection matrix.
I have established this:
If you reflect twice of across the same axis twice $Q^2 = I$.
First, it says to compute $Qu$ and simplify as much as possible. Does this just mean move the equation around to get $Qu$?
Suppose $v$ is orthogonal to $u$. Compute $Qv$. Im not sure how this is done.
Then im asked to explain in plain English which subspace $Q$ is reflecting across.
The last one states:
Compute the reflection matrix $Q_1 = I - 2u_1 u_1^T$ where $u_1 = (0,1)$. Compute $Q_1 x_1$, where $x_1 = (0,1)$ and sketch the vectors $u1$, $x1$ and $Q_1x_1$ in the plane.
I'm not sure how to even start that one.
If anyone could clear this up for me that would be much help !