Linear Combination of Vectors

A *linear combination* of vectors is a *sum of scalar
multiples* of those vectors. That is, given a set of
vectors
of the same type,^{5.4} such as
(they must have the
same number of elements so they can be added), a linear combination is
formed by multiplying each vector by a scalar
and summing
to produce a new vector
of the same type:

For example, let , , , and . Then the linear combination of and is given by

In signal processing, we think of a linear combination as a
*signal mix*. Thus, the output of a *mixing console*
may be regarded as a *linear combination* of the input signal
tracks.

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