### 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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Gram-Schmidt Orthogonalization

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General Conditions