Let be an invertible matrix. Define Prove that is an isomorphism.

From the answer provided, it shows:

Let

where do the values in blue come from?

Printable View

- Apr 22nd 2008, 10:33 PMlllllclearification on Isomorphism question
Let be an invertible matrix. Define Prove that is an isomorphism.

From the answer provided, it shows:

Let

where do the values in blue come from? - Apr 22nd 2008, 10:48 PMIsomorphism
- Apr 22nd 2008, 11:09 PMlllll
The first thing that was shown was was a linear

was shown to be 1-1 by:

was shown to be onto by:

for any Find an to show that

then

That's everything I have shown. - Apr 22nd 2008, 11:15 PMIsomorphism
On second thought, I think I will just prove it my way. Probably the book has done something similar.

Proving is a homomorphism is not hard. Proving 1-1 is even easier! So I dont understand why the textbook has done the mentioned manipulations...

If

To prove homomorphism: Let W = XY, the let us prove .

- Apr 22nd 2008, 11:20 PMIsomorphism
- Apr 22nd 2008, 11:32 PMlllll
Is there a way that this can be shown without using determinants? Other then solving numerical problems, we haven't looked at theorems governing them.

- Apr 23rd 2008, 12:10 AMIsomorphism
- Apr 23rd 2008, 09:41 AMThePerfectHacker
There is another way to show the map is onto. A consequence of the rank nullity theorem says that if you have a one-to-one linear transformation and the dimension of V and U are both equal and finite then the map is also onto. In this case which have the same finite dimension.