Invertible linear operator

[hkus10]

 

Let P belongs to Mnn be a nonsingular matrix and Let L:Mnn>>>Mnn be given by L(A) = P^-1AP for all A in Mnn. Prove that L is an invertible linear operator.

I have no clues how to start this question.

What do I need to prove for this question? and why

 

[DrRocket]

Let P belongs to Mnn be a nonsingular matrix and Let L:Mnn>>>Mnn be given by L(A) = P^-1AP for all A in Mnn. Prove that L is an invertible linear operator.

I have no clues how to start this question.

What do I need to prove for this question? and why

 

 

 

You need to prove that P^-1AP is invertible. Because that is the assignment.

 

There are several ways to prove this, in very few steps. Which you choose is a matter of taste and what you know about operators on finite-dimensional vector spaces at this point in your class.

 

1. Find the inverse.

 

2. Show that the determinant is not zero.

 

3. Show that the kernel is trivial.

 

4. Show that the rank is n.

[Xerxes]

DrRocket, ajb, me: Why bother with this guy? This is the second question almost identical to the first. We have all tried to help in our own way, maybe others have too, but I forget.

 

And what do we get for our trouble? A "thank you, I get it now"? (thanks are always welcome!). I didn't see one

 

Perhaps a "thanks but I don't quite follow - please explain" would be even better. Or maybe (s)he decided one, some or all of us were wrong. It would have been nice to know.

 

But I seriously doubt there will be any more follow-up here than in the previous thread.

 

Please slap my wrist if I ever try to help out in this sub again.

 

Supplicants here are mostly leaches - they suck your knowledge and give ziltch back, not even a "yummy, nice blood there fella"

[DrRocket]

DrRocket, ajb, me: Why bother with this guy? This is the second question almost identical to the first. We have all tried to help in our own way, maybe others have too, but I forget.

 

And what do we get for our trouble? A "thank you, I get it now"? (thanks are always welcome!). I didn't see one

 

Perhaps a "thanks but I don't quite follow - please explain" would be even better. Or maybe (s)he decided one, some or all of us were wrong. It would have been nice to know.

 

But I seriously doubt there will be any more follow-up here than in the previous thread.

 

Please slap my wrist if I ever try to help out in this sub again.

 

Supplicants here are mostly leaches - they suck your knowledge and give ziltch back, not even a "yummy, nice blood there fella"

 

Ah, maybe it will help a lurker.

 

I don't expect to get anything back. The sort of students who are typically appreciative ask deeper questions, or none at all outside of their class.

 

It's hard to get excited about something that one can solve four different ways without thinking very hard. If he/she/it decided that I was wrong it might be interesting to see what was considered right.

 

What intrigues me is that there seems to be a supply of people taking courses requiring proofs who apparently are clueless regarding the most basic subject matter. And it is May, so they have been in the class for quite a while.

[Xerxes]

Ah, maybe it will help a lurker.

 

Then I have a more Stalinist view than you. If what you call a lurker is unwilling to join this, or any other forum, as in, ask questions, seek clarification etc, then they are fully entitled to all the mis-information that abounds on these fora. And more fools they

 

What intrigues me is that there seems to be a supply of people taking courses requiring proofs who apparently are clueless regarding the most basic subject matter.

 

For "intrigues" read "depresses" and I would agree. I have spent most of my adult life arguing against the proposition that here in the UK educational standards have fallen over time, hence more students get higher grades.

 

I now freely confess I was wrong - students feel entitled to passing grades, funding bodies require a high percentage of passing grades for each dept; result - falling standards due to pressure from both sides.

 

I remember in my first year of lecturing I failed two thirds of scripts, and was told rather firmly by our Chair that I couldn't do that as it would essentially impoverish our department. "Please re-evaluate". What? Make something wrong into a half-right?

 

Needless to say I did, toady that I am. Is disillusioned the word?

[DrRocket]

Then I have a more Stalinist view than you. If what you call a lurker is unwilling to join this, or any other forum, as in, ask questions, seek clarification etc, then they are fully entitled to all the mis-information that abounds on these fora. And more fools they

 

 

 

For "intrigues" read "depresses" and I would agree. I have spent most of my adult life arguing against the proposition that here in the UK educational standards have fallen over time, hence more students get higher grades.

 

I now freely confess I was wrong - students feel entitled to passing grades, funding bodies require a high percentage of passing grades for each dept; result - falling standards due to pressure from both sides.

 

I remember in my first year of lecturing I failed two thirds of scripts, and was told rather firmly by our Chair that I couldn't do that as it would essentially impoverish our department. "Please re-evaluate". What? Make something wrong into a half-right?

 

Needless to say I did, toady that I am. Is disillusioned the word?

 

Standards have indeed fallen.

 

I have not lectured in quite a while, but I have confirmed that in at least two graduate mathematics programs at major universities new grad students are no longer prepared to take measure theory. This astonishes me, since I took such a class with essentially no pre-requisites.

 

I am also told that students are no longer willing to take notes and demand either a textbook or notes provided by the professor. I my day we worked quite hard, but did no demanding of trivia. Quite often no text existed -- anywhere -- and often we, the students, presented the bulk of the material.

 

Grades seem to be up, but accomplishment down. Since the gene pool has not changed much I conclude that the problem lies with educators and the public at large lacking backbone. We had standards, and not everyone passed. In fact in the PhD program, relatively few made it all the way. Three or four of us passed the general exam in the year that I took it. We were the first to pass it in about five years.

 

Even in high schooleople were held back. The fullback got a standing ovation when he received his diploma -- finally at about age 21.

 

I once received a call from what I later learned was the "Dean in charge of football players". We had a short conversation:

 

He: I understand that "S" was in your class. (S had been a high school All American.)

 

Me: Yes

 

He: How did he do ?

 

Me: Flunked

 

He: Why ?

 

Me: Not enough points.

 

That ended the conversation.