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Linear Functions

Linear functions have two properties of Additivity and Homogeneity:

$f(x+y) = f(x) + f(y)$

$f(\alpha x) = \alpha f(x)$, where $\alpha$ is a scalar.

System of Linear Equations

Let’s say we want to predict the power output of a wind turbine. We’ve collected data for the following:

• Feature: Feature is the “input”, in our case, wind speed.
• Target: Target is the “output”, in our case, “power output”. Given a dataset, we would want to “predict” the target using a machine learning model.

When we plot the dataset, below is how it looks:

We can see that the data points fit across a line: the goal of Linear Regression is to find the line for the best fit for the above data points.

Linear Regression assumes that the relation between input (wind speed) and output (power output) is linear (i.e. can be modeled by a straight line), and treats the dataset as a system of linear equations. Its goal is to find the values of $(m,b)$.

In Machine Learning, a straight line is generally written as:

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$y = wx + b$, where $w$ is the weight and $b$ is the bias term.

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