Linear regression is a fundamental supervised learning algorithm used for predicting a continuous target variable based on one or more predictor variables. It works by finding the best-fitting straight line (or hyperplane in higher dimensions) through the data points.

Simple linear regression involves a single independent variable (predictor) to predict a single dependent variable (target). The relationship is modeled by the equation: ( y = \beta_0 + \beta_1 x + \epsilon ), where ( y ) is the dependent variable, ( x ) is the independent variable, ( \beta_0 ) is the y-intercept, ( \beta_1 ) is the slope (coefficient), and ( \epsilon ) is the error term.

Multiple linear regression extends simple linear regression by incorporating two or more independent variables to predict the dependent variable. The equation becomes: ( y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + \epsilon ), where ( x_1, x_2, ..., x_k ) are the ( k ) independent variables and ( \beta_1, \beta_2, ..., \beta_k ) are their respective coefficients.

The estimation of coefficients in multiple linear regression also uses the OLS method, but it is typically solved using matrix algebra, which is more efficient for handling multiple variables. The model aims to find the hyperplane that best fits the data in a multi-dimensional space.

When using linear regression, it's important to consider assumptions such as linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of errors. Violations of these assumptions can affect the reliability of the model's predictions and inferences. Feature scaling might be necessary for some algorithms, but for standard OLS linear regression, it's not strictly required for coefficient estimation, though it can be beneficial for regularization techniques.