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Apr13-11, 11:42 PM
A "dramatic" use of the use of a basis in a vector space is expressing a function as a sum of other functions. For example, fourier series, Chebeshev polynomials.
Careful, you need
for that. On the topic of Fourier, each piano key corresponds to a sine function. The collection of finite sums gives the set of all piano chords.
There is also
Lattice based crypto
, but that is discrete so ymmv.