### Changing Coordinates

What's more interesting is when we project a signal onto a set
of vectors *other than* the coordinate set. This can be viewed
as a *change of coordinates* in . In the case of the DFT,
the new vectors will be chosen to be *sampled complex sinusoids*.

#### An Example of Changing Coordinates in 2D

As a simple example, let's pick the following pair of new coordinate vectors in 2D:

These happen to be the DFT sinusoids for having frequencies
(``dc'') and (half the sampling rate). (The sampled complex
sinusoids of the DFT reduce to real numbers only for and .) We
already showed in an earlier example that these vectors are *orthogonal*. However, they are not orthonormal since the norm is
in each case. Let's try projecting onto these vectors and
seeing if we can reconstruct by summing the projections.

The projection of onto is, by
definition,^{5.12}

Similarly, the projection of onto is

The sum of these projections is then

It worked!

**Next Section:**

Projection onto Linearly Dependent Vectors

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Projection