linear operator and reflection

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October 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.

Thanks to all for any help you give.
October 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.
October 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 $T: R^2 \to R^2$ 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.
October 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
October 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.