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Change of Orthonormal Basis Worksheet In the change of basis worksheet, we saw that a pair of bases (v1 , . , vn ) and (v1 , . , vn ) for a vector...

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Change of Orthonormal Basis Worksheet In the change of basis worksheet, we saw that a pair of bases ( v 1 ,...,v n ) and ( v 0 1 ,...,v 0 n ) for a vector space V could be related by a invertible change of basis matrix P defined by ( v 0 1 ,...,v 0 n ) = ( v 1 ,...,v n ) P . The columns of P were calculated by computing the component of the vectors ( v 0 1 ,...,v 0 n ) in the basis ( v 1 ,...,v n ). Also if M is the matrix of f end( V ) in the basis ( v 1 ,...,v n ), its matrix M 0 in the basis ( v 0 1 ,...,v 0 n ) is given by M 0 = P - 1 MP . If you are lucky enough to have orthonormal bases, life is easier. .... For example, consider C 3 with standard inner product h ( z 1 ,z 2 ,z 3 ) , ( w 1 ,w 2 ,w 3 ) i = 3 i =1 ¯ w i z i . An orthonormal basis is f 1 = (1 ,i, 0) / 2 ,f 2 = (1 , - i, 0) / 2 ,f 3 = (0 , 0 , 1). To compute the components of a vector v in this basis, you only have to compute h v,f 1 i , h v,f 2 i , h v,f 3 i . Calculate the components of the vector v = (1 , 1 , 1) in the basis ( f 1 ,f 2 ,f 3 ) using the inner product. Now let e 1 = (1 , 0 , 0) ,e 2 = (0 , 1 , 0) ,e 3 = (0 , 0 , 1) be the canonical (orthonormal) basis for C 3 . Calculate the change of basis matrix P from the basis ( e 1 ,e 2 ,e 3 ) to ( f 1 ,f 2 ,f 3 ). Remember that P is obtained from P by taking the transpose and complex conjugate. Compute P and P P . What can you say about P ? EXPLAIN YOUR OBSERVATION! 1
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