Consider this example:

These point in different directions, but they are not orthogonal. What happens now? The projections are

The sum of the projections is

So, even though the vectors are linearly independent, the sum of
projections onto them does not reconstruct the original vector. Since the
sum of projections worked in the orthogonal case, and since orthogonality
implies linear independence, we might conjecture at this point that the sum
of projections onto a set of
vectors will reconstruct the original
vector only when the vector set is *orthogonal*, and this is true,
as we will show.

It turns out that one can apply an orthogonalizing process, called
*Gram-Schmidt orthogonalization* to any
linearly independent
vectors in
so as to form an orthogonal set which will always
work. This will be derived in Section 5.10.4.

Obviously, there must be at least
vectors in the set. Otherwise,
there would be too few *degrees of freedom* to represent an
arbitrary
. That is, given the
coordinates
of
(which are scale factors relative to
the coordinate vectors
in
), we have to find at least
coefficients of projection (which we may think of as coordinates
relative to new coordinate vectors
). If we compute only
coefficients, then we would be mapping a set of
complex numbers to
numbers. Such a mapping cannot be invertible in general. It
also turns out
linearly independent vectors is always sufficient.
The next section will summarize the general results along these lines.

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