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.