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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

*Last edited by Stanley_Marsh (2007-04-17 09:13:16)*

Numbers are the essence of the Universe

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

I did

I must show that R(H)=R(F) ..don't know what to do then.

Numbers are the essence of the Universe

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

You know how to show that two functions are equal? Two functions φ and ψ (having the same domain) are equal iff φ(*x*) = ψ(*x*) for all *x* in the domain.

*Last edited by JaneFairfax (2007-04-17 10:20:35)*

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Jane's proof may seem at first to just be pushing things around. But think about the reason underlying it. We have a maps F, G, and H. We can either compose H and G to create a new map first, and then compose this with F, or we can compose G and F first, and then compose this we H. Does it matter what we do first? Will it really change where things get sent? In short, no, it doesn't matter. Sending an element through one function at a time, versus sending it through the bijection, it all ends up in the same place. It's this fact that Jane's proof uses.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

I think I've got it , Thx guys~

Numbers are the essence of the Universe

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,313

No, Jane provided the Standard reply to this question, which you will see in any textbook.

**X'(y-Xβ)=0**

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