Now consider another example:

The projections of onto these vectors are

The sum of the projections is

Something went wrong, but what? It turns out that a set of
vectors can be used to reconstruct an arbitrary vector in
from
its projections only if they are *linearly independent*. In
general, a set of vectors is linearly independent if none of them can
be expressed as a linear combination of the others in the set. What
this means intuitively is that they must ``point in different
directions'' in
-space. In this example
so that they
lie along the *same line* in
-space. As a result, they are
linearly *dependent*: one is a linear combination of the other
(
).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University