# Invariance of Transformations

1. Oct 12, 2009

### Hallingrad

Hey guys,

I was wondering how you would go about proving that the image of a transformation T, im(T), is invariant? And following that, how would you prove T(W1 $$\bigcap$$ W2) is invariant if T(W1) and T(W2) are both invariant.

On an unrelated note, another questions asks to show that
TX = X - (P^-1 * X * P) is a linear operation, but no matter what I do, I always come up with it showing that it's in fact not a linear operation. What do you guys think?

Thanks a lot for any help ^_^.

2. Oct 12, 2009

### Office_Shredder

Staff Emeritus
Proving the image of a transformation is invariant is pretty straightforward from the definition. Where are you getting stuck?

3. Oct 12, 2009

### Hallingrad

Well we haven't yet covered what an invariant transformation is, but the assignment is due before the next lecture :/. Any advice on where to start?

4. Oct 12, 2009

### Office_Shredder

Staff Emeritus
A subspace W of a vector space V is called invariant under T if T(W) is a subset of W (so T can be defined as a function from W to W also). Invariance isn't describing the function, it's describing the subspace.

5. Oct 12, 2009

### Hallingrad

With that in mind though, how do you show that T(im(T) = im(T)? That is what we're trying to show, right? And what about the intersection of W1 and W2?

6. Oct 12, 2009

### Office_Shredder

Staff Emeritus
You want to show that T(im(T)) is a subset of im(T). So if x is in T(im(T)) then x is in im(T). Can you see why this is true?

7. Oct 13, 2009

### Hallingrad

I have a loose grasp on why that may be, but what's the formal proof?

8. Oct 16, 2009

### Hallingrad

After another week of classes, he still hasn't addressed invariant transformations. I understand that the image is to the column space as the kernel is to the nullspace for a transformations. That is, the image is what is the output of a given transformation. Yet I still don't see how T(im(T)) = im(T). The same goes for showing that the intersection of two spaces is also T invariant. Any help would be much appreciated!

9. Oct 17, 2009

### Office_Shredder

Staff Emeritus
You don't want that T(im(T)) = im(T), you just want that it is a subset! Consider the rewording of the question: Show that any element in im(T), say y, has the property that T(y) is in im(T). Do you see why this is always true?

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