Is the result of a matrix transformation equivalent to the that of that same matrix but orthonormali

pouzdrotf 2022-07-06 Answered
Is the result of a matrix transformation equivalent to the that of that same matrix but orthonormalized?
Say we have the transformation matrix A of full rank such that A 1 A t ,, i.e., the matrix A consists of linearly independent vectors which aren't orthogonal to each other, a vector v, and the orthonormalized transformation matrix A . Is it true that A v = A v ? And if not, is this unimportant?
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Answers (1)

Karla Hull
Answered 2022-07-07 Author has 20 answers
One of the key features of the Gram-Schmidt process is that the vectors generated by it span the same k-vector subspace as the original vectors. Put into the language of matrices, this means that the column space of A (or the range of T) is the same as the column space of A (or the range of T ). (Here, I assume that the linearly independent vectors we started with were the columns of A ..) Consequently, we have that
{ w V | T ( v ) = w  for some  v V } = { w V | T ( v ) = w  for some  v V } .
But that does not imply that A v = T ( v ) = T ( v ) = A v for all v V ..
For instance, consider the R -vector space R 2 , i.e., the set of column vectors ( a , b ) t for a , b R . Observe that the vectors v = ( 1 , 1 ) t and w = ( 1 , 2 ) t are linearly independent (because the matrix with columns v and w has determinant 1). By Gram-Schmidt, we obtain an orthogonal basis v 1 = ( 1 , 1 ) t and v 2 = ( 1 2 , 1 2 ) t , and we can convert this into an orthonormal basis by taking v = v 1 | | v 1 | | and w = v 2 | | v 2 | | .
Using these as columns, we can form the 2 × 2 matrix A . We will denote by e 1 = ( 1 , 0 ) and e 2 = ( 0 , 1 ) the standard basis vectors of R 2 . Observe that we have
A e 1 = v v 1 | | v 1 | | = v = A e 1 ,
and the analogous statement holds for e 2 in place of e 1 .
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