Orthogonal transformations

daniel_i_l

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.

StatusX

Yes, and it's true more generally for any invertible transformation.