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:
![$\displaystyle \underline{y}= \alpha_1 \underline{x}_1 + \alpha_2 \underline{x}_2 + \cdots + \alpha_M \underline{x}_M
$](http://www.dsprelated.com/josimages_new/mdft/img715.png)
![$ \underline{x}_1=(1,2,3)$](http://www.dsprelated.com/josimages_new/mdft/img716.png)
![$ \underline{x}_2=(4,5,6)$](http://www.dsprelated.com/josimages_new/mdft/img717.png)
![$ \alpha_1=2$](http://www.dsprelated.com/josimages_new/mdft/img718.png)
![$ \alpha_2=3$](http://www.dsprelated.com/josimages_new/mdft/img719.png)
![$ \underline{x}_1$](http://www.dsprelated.com/josimages_new/mdft/img720.png)
![$ \underline{x}_2$](http://www.dsprelated.com/josimages_new/mdft/img721.png)
![$\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).
$](http://www.dsprelated.com/josimages_new/mdft/img722.png)
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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Linear Vector Space
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Scalar Multiplication