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.
Next Section: General ConditionsPrevious Section: Projection onto Linearly Dependent Vectors