anhilators of vector spaces.

I am trying to prove that

I have managed to prove one inclusion

The fact that any function in these spaces is a linear transformation was really helpful however if doesn't seem to really help me on the reverse inclusion.

Let

then and

since is a linear transformation:

thus and

I dont think this helps me. I can't necessarily say and

What should I do from here? Is this the right path and I'm just not catching something?

Re: anhilators of vector spaces.

Quote:

Originally Posted by

**Jame** I am trying to prove that

I have managed to prove one inclusion

The fact that any function in these spaces is a linear transformation was really helpful however if doesn't seem to really help me on the reverse inclusion.

Let

then

and

since

is a linear transformation:

thus

and

I dont think this helps me. I can't necessarily say

and

What should I do from here? Is this the right path and I'm just not catching something?

Ok, so let then we know that for all , right? But, isn't ?

Re: anhilators of vector spaces.

Oh i see

so

and

Particularly when , and since f is an L.T.,

therefore

implies

similarly for

so

Thanks a lot!

Re: anhilators of vector spaces.

Sorry, I have another question having to do with this particular equality.

I've read you can use this to prove that

by asserting that they have the same dimension.

I have been playing around with this for the past hour or so. And haven't really gotten anywhere.

I've got

but I have found no equalities that I could apply to.

Re: anhilators of vector spaces.

Quote:

Originally Posted by

**Jame** Sorry, I have another question having to do with this particular equality.

I've read you can use this to prove that

by asserting that they have the same dimension.

I have been playing around with this for the past hour or so. And haven't really gotten anywhere.

I've got

but I have found no equalities that I could apply

to.

Which part are you trying to prove. I think you've proven the dimension formula and are trying to see how to use it. Well, as I mentioned earlier it's trivial to prove that but since they have the same dimension, they must be equal. Is that what you;re after?

Re: anhilators of vector spaces.

Proving

This is just an application if the equality in the very first post? How does this work?

Re: anhilators of vector spaces.

Re: anhilators of vector spaces.

i'm trying to follow your reasoning, and it looks circular.

how does

without first showing what we are trying to prove, namely:

?

Re: anhilators of vector spaces.

Quote:

Originally Posted by

**Deveno** i'm trying to follow your reasoning, and it looks circular.

how does

without first showing what we are trying to prove, namely:

?

Perhaps I mixed up what we are trying to prove with what we're proving. But, we first proved (through the first three posts) that which is what I used. I'm tired and don't feel like making sure I didn't mix them up, which I did initially. So, if I did, which I don't think I did, feel free to do it up man.

Re: anhilators of vector spaces.

from the first 3 posts, we know that they are equal (the annihilator of the sum, and the intersection of the annihilators). so they have to have the same dimension.

(assuming, of course, that their dimension is finite).

i may be mistaken, but i am wondering if what the thread starter is asking is, can we show equality from a dimensional argument?

i think we can, but we're going to have to invoke a basis, at some point.

(was your "non-obvious" assertion ?)

Re: anhilators of vector spaces.

Quote:

Originally Posted by

**Deveno** from the first 3 posts, we know that they are equal (the annihilator of the sum, and the intersection of the annihilators). so they have to have the same dimension.

(assuming, of course, that their dimension is finite).

i may be mistaken, but i am wondering if what the thread starter is asking is, can we show equality from a dimensional argument?

i think we can, but we're going to have to invoke a basis, at some point.

Ok, now I wonder if you realize that he asked to prove a DIFFERENT equality than the one initially asked...I originally proved "the annihilator of a sum is the interesction of annihilators" and the post that you thought I had circular reasoning on I was proving "the sum of annihilators is the annihilator of the intersection" for which I used "the annihilator of a sum is the interesction of annihilators"

Re: anhilators of vector spaces.

ok, i got it. yes, the similarity DID confuse me. and in the case we're talking about:

Ann(W1+W2) = Ann(W1)∩Ann(W2) DOES imply Ann(W1∩W2) = Ann(W1)+Ann(W2)

for a moment, there, we were talking about our conversation, and i got "meta-lost".