Employing non-full-rank design matrices throughout, this text provides a concise yet solid foundation for understanding basic linear models. It introduces the basic algebra and geometry of the linear least squares problem, before delving into estimability and the GaussMarkov model. After presenting the statistical tools of hypothesis tests and confidence intervals, the author analyzes mixed models, such as two-way mixed ANOVA, and the multivariate linear model. The text presents proofs and discussions from both algebraic and geometric viewpoints and includes exercises of varying levels of difficulty at the end of each chapter.