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I'm just here to ask if there is anything dramatic or marginally more complex when you project a vector onto a subspace that is not defined by an orthonormal basis. We are currently studying an elementary course, and the strict definition of projections onto subspaces emphasizes the fact that the basis of the subspace is orthonormal. What could go wrong otherwise?

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    $\begingroup$ Naturally the subspace doesn't change whether you use an orthonormal basis or not. The computation of the orthogonal projection may be more complicated, however, than the convenient case of using an orthonormal basis. $\endgroup$ – Muphrid Dec 13 '15 at 1:51
  • $\begingroup$ In an inner-product space the orthogonal projection of a vector v onto a closed linear subspace S is not dependent on choice of basis for S. It is a function of v and S. If however you compute it by applying the computational formula that uses an orthonormal basis for S, to a basis for S that is not orthonormal,the chances of being right are between slim and none. $\endgroup$ – DanielWainfleet Dec 13 '15 at 5:43
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If the basis of your subspace is just orthogonal (not normalized) then nothing major changes: you just have to divide each term by the norm squared of the basis vector. However if your basis is not orthogonal even then you've got two options:

  1. Orthogonalize your basis vectors via the Gram-Schmidt process and then proceed as normal
  2. Find the dual basis for that subspace

Each of the above is a bit more complex (but just a little bit) than simply using the projection formula. If you want more info, just let me know in a comment below.

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