### Signal/Vector Reconstruction from Projections

We now arrive finally at the main desired result for this section:**Theorem:**The projections of any vector onto any orthogonal basis set for can be summed to reconstruct exactly.

*Proof:*Let denote any orthogonal basis set for . Then since is in the space spanned by these vectors, we have

for some (unique) scalars . The projection of onto is equal to

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