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 Post subject: Basis ProblemPosted: Fri, 1 Apr 2011 20:03:34 UTC
 S.O.S. Newbie

Joined: Fri, 1 Apr 2011 19:49:14 UTC
Posts: 3
Hello everyone.

Let be a 3-D space over a field with basis and consider linear operators whose matrices relative to are:

Find vectors such that

with
with

I have been asked to find these vectors so that the matrices of with respect to are stricly upper triangular.

My inital procedure was to first find a common eigenvector for and take that as , but I couldn't find a common eigenvector that worked, when I calculated , the respective matrices for were not stricly upper triangular. I then tried but that didn't work. Any help would greatly appreciated.

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 Post subject: Re: Basis ProblemPosted: Sat, 2 Apr 2011 05:57:26 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Patient wrote:
Hello everyone.

Let be a 3-D space over a field with basis and consider linear operators whose matrices relative to are:

Find vectors such that

with
with

I have been asked to find these vectors so that the matrices of with respect to are stricly upper triangular.

My inital procedure was to first find a common eigenvector for and take that as , but I couldn't find a common eigenvector that worked, when I calculated , the respective matrices for were not stricly upper triangular. I then tried but that didn't work. Any help would greatly appreciated.

What do you mean by " doesn't work"?

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 Post subject: Posted: Sat, 2 Apr 2011 14:01:39 UTC
 S.O.S. Newbie

Joined: Fri, 1 Apr 2011 19:49:14 UTC
Posts: 3
Taking gives but then letting and with gives for some . for some . for some . This doesn't really get me anywhere to finding out what is.

I tried to work out similarly. This method doesn't give so that the respective matrices are upper triangular. BUt that's probably because I'm going about it in entirely the wrong way.

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 Post subject: Posted: Sat, 2 Apr 2011 14:46:11 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Patient wrote:
Taking gives but then letting and with gives for some . for some . for some . This doesn't really get me anywhere to finding out what is.

I tried to work out similarly. This method doesn't give so that the respective matrices are upper triangular. BUt that's probably because I'm going about it in entirely the wrong way.

It is the right method (modulo the fact that you are expressing these vectors with respect to the basis B without saying so in your posts), and you should have some idea what must be like. Of course you cannot determine uniquely what must be, because you can scale any basis element, as well as adding multiples of which is in the kernel.

So from you know a-c=0, so . Indeed any and any would do. Can you see why?

And you can choose almost freely --- the only constraint is that . Can you see why?

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 Post subject: Posted: Sat, 2 Apr 2011 15:10:39 UTC
 S.O.S. Newbie

Joined: Fri, 1 Apr 2011 19:49:14 UTC
Posts: 3
I think I understand, I previously calculated the same way, so taking as in your post, and taking such that .

My problem comes when I try to calculate the matrices of the linear operators with respect to . I can't understand how these matrices become stricly upper triangular. Another problem I hadn't previously stated was that I am supposed to write down an explicit change of basis matrix between and , which I don't know how to do with such arbitrary and without knowing .

Thanks for your responses, I appreciate the help.

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