Register to reply

Orthogonal transformations

by daniel_i_l
Tags: orthogonal, transformations
Share this thread:
daniel_i_l
#1
Jul19-07, 05:48 AM
PF Gold
daniel_i_l's Avatar
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.
Phys.Org News Partner Science news on Phys.org
Apple to unveil 'iWatch' on September 9
NASA deep-space rocket, SLS, to launch in 2018
Study examines 13,000-year-old nanodiamonds from multiple locations across three continents
StatusX
#2
Jul19-07, 06:28 AM
HW Helper
P: 2,567
Yes, and it's true more generally for any invertible transformation.


Register to reply

Related Discussions
Gauge Transformations and (Generalized) Bogoliubov Transformations. Quantum Physics 8
Regarding Orthogonal Transformations Linear & Abstract Algebra 11
Orthogonal transformations Calculus & Beyond Homework 2
Orthogonal equation plane help Introductory Physics Homework 3
Find a vector orthogonal Introductory Physics Homework 5