Proving Invariance of Transformations and the Linearity of a Specific Operation

[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 ^_^.

 

[Office_Shredder]

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

 

[Hallingrad]

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

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?

 

[Office_Shredder]

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.

 

[Hallingrad]

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.

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?

 

[Office_Shredder]

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?

 

[Hallingrad]

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?

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

 

[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!

 

[Office_Shredder]

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?