Math Help - linear operator and reflection

1. linear operator and reflection

Morning all, I am having difficulties doing these two question so if you could be kind enough to offer me some help it would be greatly appreciated.

1,Prove that a linear operator on R^2 is a reflection iff it's eigenvalues are -1 and 1, and the eigenvectors associated with these eigenvalues are orthogonal.

2 Prove that a conjugate of a glide reflection in M is a glide reflection and prove that the glide vectors have the same length.

2. Re: linear operator and reflection

ok, so I have worked out the first question but I still can't get the second one.

Thank you for any help.

3. Re: linear operator and reflection

Well I know how to do q1, so I will just make sure you have worked it out right.

Sorry, I can't help with q2 but I am sure someone will help you soon.

First, if $T: R^2 \to R^2$ is a reflection along the line $m$, we can then let $u$ be a vector along this line and let $v$ be a vector in $R^2$which is orthogonal to it. We can then say that $\{u,v\}$ form a basis for $R^2$ and $T(u)=u$ and $T(v)=-v$ so the eigenvalues for $T$ are $1$ and $-1$. We can also say that for $a,b \in R^2$, the eigenvectors are $av$ and $bu$ which are obviously orthogonal by construction.
You then need to show the converse, but that is easy enough.

Sorry again about not being able to help for q2.

4. Re: linear operator and reflection

Thanks shelford, yeah that's is pretty much how I did it so thanks.

But I am still unsure how to do question 2, so any help guys would be very nice.

Thanks

5. Re: linear operator and reflection

Originally Posted by philistine
Thanks shelford, yeah that's is pretty much how I did it so thanks.

But I am still unsure how to do question 2, so any help guys would be very nice.

Thanks
Your welcome, I have thought about q2 but I can't get it.

Someone else will have an idea about it and help.