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Mathematics of the DFT
    Geometric Signal Theory
       Linear Combination of Vectors


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Linear Combination of Vectors

A linear combination of vectors is a sum of scalar multiples of those vectors. That is, given a set of $ M$ vectors $ \underline{x}_i$ of the same type,5.4 such as $ {\bf R}^N$ (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 $ \alpha_i$ and summing to produce a new vector $ \underline{y}$ of the same type:

$\displaystyle \underline{y}= \alpha_1 \underline{x}_1 + \alpha_2 \underline{x}_2 + \cdots + \alpha_M \underline{x}_M $

For example, let $ \underline{x}_1=(1,2,3)$, $ \underline{x}_2=(4,5,6)$, $ \alpha_1=2$, and $ \alpha_2=3$. Then the linear combination of $ \underline{x}_1$ and $ \underline{x}_2$ is given by

$\displaystyle \underline{y}= \alpha_1\underline{x}_1 + \alpha_2\underline{x}_2 = 2\cdot(1,2,3) + 3\cdot(4,5,6) = (2,4,6)+(12,15,18) = (14,19,24). $

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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