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#1 2014-07-13 10:52:30

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Invertible matrix

Problem:
Let

be a square invertible matrix. Let
be the linear transformation defined by
. Prove that
is an isomorphism.

How to start the proof?

I am thinking about something like this: Let

, then by definition of matrix invertibility
. So
.

But there is an issue of order, it does matter in the matrix product and I do violate it in this proof. So it must be wrong. But I do not see how to fix it.

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#2 2014-07-13 11:02:21

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Invertible matrix

Well, first you need to prove it's a homomorphism. Do you know what that means and how to do that?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2014-07-13 14:27:57

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Invertible matrix

Homomorphism? I guess not. At least what I read in my lecture notes does not give me any clues on how to solve the problem.

Ok. Taking a step back. According to my notes:
- Let T:V->W be a linear transformation.
- If T is invertible, we call it an isomorphism.

So to prove that

is isomorphic, I need to show that it is invertible. Yes?
And if T is invertible, that  would mean that
has a unique solution.

Well, maybe the trick is in the properties of invertible matrices?


It does look kinda nice...
Does that work as a proof, or did I prove something else?

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#4 2014-07-13 14:38:11

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Invertible matrix

Oops. mistake.


So my nice set of equations is not correct sad

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#5 2014-07-13 22:34:17

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Invertible matrix

Well, the inverse transformation is T'(A)=BAB^-1


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2014-07-14 06:38:05

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Invertible matrix


hmm... Ok. It does look correct.

The next question is what if A is not invertible? Does it affect invertibility of T or not?

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#7 2014-07-14 06:57:20

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Invertible matrix

What are x and y?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#8 2014-07-14 07:54:51

White_Owl
Member
Registered: 2010-03-03
Posts: 99

Re: Invertible matrix

Vectors which undergoing the transformation.

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#9 2014-07-14 08:54:58

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Invertible matrix

Isn't the transformation done on a matrix?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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