Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

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

Problem:

Let

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.

Offline

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

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

Offline

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

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?

Offline

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

Oops. mistake.

So my nice set of equations is not correct

Offline

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

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

Offline

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

hmm... Ok. It does look correct.

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

Offline

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

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

Offline

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

Vectors which undergoing the transformation.

Offline

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

Isn't the transformation done on a matrix?

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

Offline

Pages: **1**