# linear operator and reflection

• Oct 12th 2011, 04:53 AM
philistine
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.

• Oct 13th 2011, 06:04 AM
philistine
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.
• Oct 14th 2011, 04:13 AM
shelford
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 $\displaystyle T: R^2 \to R^2$ is a reflection along the line $\displaystyle m$, we can then let $\displaystyle u$ be a vector along this line and let $\displaystyle v$ be a vector in $\displaystyle R^2$which is orthogonal to it. We can then say that $\displaystyle \{u,v\}$ form a basis for $\displaystyle R^2$ and $\displaystyle T(u)=u$ and $\displaystyle T(v)=-v$ so the eigenvalues for $\displaystyle T$ are $\displaystyle 1$ and $\displaystyle -1$. We can also say that for $\displaystyle a,b \in R^2$, the eigenvectors are $\displaystyle av$ and $\displaystyle bu$ 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.
• Oct 15th 2011, 03:48 AM
philistine
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
• Oct 15th 2011, 03:46 PM
shelford
Re: linear operator and reflection
Quote:

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.