### 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:*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}

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Projection onto Linearly Dependent Vectors

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Projection