Math Is Fun Forum

White_Owl

Member

Registered: 2010-03-03

Posts: 106

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.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

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?

White_Owl

Member

Registered: 2010-03-03

Posts: 106

Re: Invertible matrix

Oops. mistake.

So my nice set of equations is not correct

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

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?

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

White_Owl

Member

Registered: 2010-03-03

Posts: 106

Re: Invertible matrix

Vectors which undergoing the transformation.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.