### Projection onto Non-Orthogonal Vectors

Consider this example:*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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General Conditions

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Projection onto Linearly Dependent Vectors