In the study of signals and signal processing, there is a particular mathematical operation that will show up quite a few times, that of a
linear combination . Given a collection of vectors in a vector space, say $M$ vectors $x_0, x_1, \dots x_{M-1} \in C^N$, and $M$ scalars $\alpha_0, \alpha_1, \dots, \alpha_{M-1} \in C$, then the linear combination of these vectors is:
$y=\alpha_0 x_0 + \alpha_1 x_1 + \dots + \alpha_{M-1} x_{M-1} = \sum_{m=0}^{M-1} \alpha_m x_m$. The result is itself also a vector in the vector space.
Linear combination example: a mixing board
A linear combination is a scaled sum of different vectors in a vector space. A real world example of a linear combination is the mixing board used in music recording studios. The board takes in a variety of different inputs and combines them--amplifying some, reducing the level of others--to create a single output of music. Mathematically, we could say $x_0=$ drums, $x_1=$ bass, $x_2=$ guitar, \dots, $x_{22}=$ saxophone, $x_{23}=$ singer, and then the mixing board creates the linear combination
$y = \alpha_0 x_0 + \alpha_1 x_1 + \dots + \alpha_{M-1} x_{M-1} = \sum_{m=0}^{M-1} \alpha_m x_m$.Changing the different $\alpha_m$'s would result in a different kind of sound that either emphasized or de-emphasizes certain instruments. For example, the producer in the studio that day may be particularly interested in the cowbell.
Linear combinations as matrix multiplication
It is possible to express a linear combination without explicitly using sums, but rather as the product of a matrix and a vector. To do this, all of the vectors to be scaled and summed in the linear combination must first be arranged into a matrix:
$\begin{align*}X&= \begin{bmatrix} x_0 | x_1 | \cdots | x_{M-1} \end{bmatrix}\\&=\begin{bmatrix}
x_0[0]&x_1[0]&\cdots&x_{M-1}[0] \\x_0[1]&x_1[1]&\cdots&x_{M-1}[1] \\\vdots&\vdots&&\vdots \\
x_0[N-1]&x_1[N-1]&\cdots&x_{M-1}[N-1] \end{bmatrix}\end{align*}$
Then the scaling factors for each vector are stacked into a single vector:$a = \begin{bmatrix}\alpha_0 \\ \alpha_1 \\ \vdots \\ \alpha_{M-1} \end{bmatrix}$
Once those are in place, the linear combination is ultimately expressed as a simple matrix-vector multiplication:$y=\alpha_0 x_0 + \alpha_1 x_1 + \dots + \alpha_{M-1} x_{M-1} = \sum_{m=0}^{M-1} \alpha_m x_m
= \begin{bmatrix}x_0 | x_1 | \cdots | x_{M-1} \end{bmatrix}\begin{bmatrix}\alpha_0 \\ \alpha_1 \\ \vdots \\ \alpha_{M-1} \end{bmatrix}= X a$
