Invariance of Transformations

1. Oct 12, 2009

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

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

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

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

8. Oct 16, 2009