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 $\mathbb{R}^N$ (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:

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

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

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

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.