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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