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Let { a 1 , a 2 , ... , a n } be an orthonormal basis of column vectors for R n , and let C be an orthogonal n x n matrix. Show that {C a 1 , C a 2 , ... , C a n } is also an orthonormal basis for R n . Solution {a1, a2, ... , an} is orthogonal basis which means ||a1||^2 = ||a2||^2 = ........= ||an||^2 = 1 and inner product of any 2 elements is 0. eg. =0 now C is orthogonal n x n matrix. {Ca1, Ca2, ... , Can} is new set. ||Ca1||^2 = |C|^2 * ||a1||^2 = 1*1 =1 similarly all others determinats are also 1. now inner product = |C|^2 * = 1*0 = 0 similarly all other inner products are also 0. hence {Ca1, Ca2, ... , Can} is also orhonormal basis. .

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- 1. Let { a 1 , a 2 , ... , a n } be an orthonormal basis of column vectors for R n , and let C be an orthogonal n x n matrix. Show that {C a 1 , C a 2 , ... , C a n } is also an orthonormal basis for R n . Solution {a1, a2, ... , an} is orthogonal basis which means ||a1||^2 = ||a2||^2 = ........= ||an||^2 = 1 and inner product of any 2 elements is 0. eg. =0 now C is orthogonal n x n matrix. {Ca1, Ca2, ... , Can} is new set. ||Ca1||^2 = |C|^2 * ||a1||^2 = 1*1 =1 similarly all others determinats are also 1. now inner product = |C|^2 * = 1*0 = 0 similarly all other inner products are also 0. hence {Ca1, Ca2, ... , Can} is also orhonormal basis.