I would appreciate a hint for a question I have been struggling with for the last couple of days.
Let $v, w \in R^2$, and assume $||v||=||w||$.
Prove that there is only one reflection matrix that fulfills $Av=w$.
I’ve been trying to use the fact that $u+v$ and $u-v$ are orthogonal because $<u+v, u-v> = 0$ since $v$ and $u$ are from the same length and then look at $A(v+w)$ but that did not lead me anywhere.
Hint: Indeed, $v+w$ and $v-w$ are orthogonal and non-zero, which means that they form a basis of $\Bbb R^2$. If it is known what a linear transformation does to each element of a basis, then the linear transformation is completely determined (i.e. there is only one possible matrix for the transformation).
Now, what should the reflection do to the vector $v+w$? What should the reflection do to $v-w$? Try to figure out how the line across which the transformation should reflect is related to $v+w$ or $v-w$.