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Orthogonal transformations 
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#1
Jun2907, 06:29 AM

PF Gold
P: 867

1. The problem statement, all variables and given/known data
Given two subspaces of R^n  W_1 and W_2 where dimW_1 = dimW_2 =/= 0. Prove that there exists an orthogonal transformation T:R^n > R^n so that T(W_1) = T(W_2) 2. Relevant equations 3. The attempt at a solution If dimW_1 = dimW_2 = m then we can say that {v_1,...,v_m} is a basis of W_1 and {u_1,...,u_m} is a basis of W_2. We can make from these bases of R^n: {v_1,...,v_m,a_1,...,a_{nm}} is a basis of R^n and {u_1,...,u_m,b_1,...,b_{nm}} is also. From these we can make orthonormal bases of R^n (via GS) so that: {v'_1,...,v'_m,a'_1,...,a'_{nm}} is an orthonormal basis of R^n and {u'_1,...,u'_m,b'_1,...,b'_{nm}} is also. Now we make a transformation where: T(v'_i) = u'_i T(a'_i) = b'_i Now, since the transformation of an orthonormal base gives another orthonormal base it's a ON transformation. And since {v'_1,...,v'_m} is a basis of W_1 and{u'_1,...,u'_m} is a basis of W_2 then T(W_1) = W_2 Is that a complete proof? Thanks. 


#2
Jun2907, 10:22 PM

Sci Advisor
HW Helper
Thanks
P: 25,251

That's pretty much it. Though you may want to point out that you've diligently applied GS in such a way as to ensure that (v'_1..v'_m) and (u'_1...u'_m) are still a basis for W_1 and W_2. I.e. what order to do GS in? You are probably already thinking of the correct order. I'm just saying this because I can't think of anything else to fault and am wondering why you bothered to post this in the first place. It is really easy.



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