Linear Systems

Now that we’re familiar with linear functions and linear combinations, let’s dive into the process of using these concepts to define real-life systems as mathematical systems.

Linear modeling

We often come across data consisting of several attributes of interest to us. For example, the records of COVID-19 patients contain many attributes worth recording, including confirmation data, fever level, blood cell counts, medicine in use, and so on. One important attribute is the survival rate, which may depend on the other attributes. If we think that there are $d$ attributes in numeric form that are represented as vectors or arrays, then a typical $i^{th}$ record looks like the following: $\bold{x_i} = \begin{bmatrix} x_{1i}\\x_{2i}\\x_{3i} \\ \vdots \\x_{di}\end{bmatrix}$. The corresponding attribute, survival, can be denoted with $y_i$ having either the values, $0$ or $1$ , corresponding to “did not survive” and “survived,” respectively. If there’s a linear relationship between $\bold{x_i}$ and $y_i$, then there exists a linear function $f$ with parameters $\bold{w} = \begin{bmatrix} w_1\\w_2\\w_3 \\ \vdots \\w_d\end{bmatrix}$ ...