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

(using the linearity of the projection operator which follows from linearity of the inner product in its first argument). Since the basis vectors are orthogonal, the projection of onto is zero for :

We therefore obtain

Therefore, the sum of projections onto the vectors , , is just the linear combination of the which forms :

by Eq.(5.3).

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