Orthogonal transformations


by daniel_i_l
Tags: orthogonal, transformations
daniel_i_l
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#1
Jul19-07, 05:48 AM
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P: 867
1. The problem statement, all variables and given/known data
I have a general question. If we have some subspace W of R^n where dimW=k. Then if T is an orthogonal transformation from R^n->R^n is the dimension of T(W) also k?


2. Relevant equations



3. The attempt at a solution

The reason I think this is true is because if {w_1,...,w_k} is an orthonormal basis of W and {w_1,...,w_k,w_(k+1),...,w_n} is an orthonormal basis of R^n then {Tw_1,...,Tw_k,Tw_(k+1),...,Tw_n} Is also an orthonomal basis of R^n. But T(W)=Sp({Tw_1,...,Tw_k}) and if {Tw_1,...,Tw_k,Tw_(k+1),...,Tw_n} is an orthonormal basis then {Tw_1,...,Tw_k} are linearly independent and dimT(W) = k.

Is this true?
Thanks.
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StatusX
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#2
Jul19-07, 06:28 AM
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P: 2,566
Yes, and it's true more generally for any invertible transformation.


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