This book offers an introduction to algorithmic-numerical thinking using basic problems of linear algebra. By focusing on linear algebra, it ensures a stronger thematic coherence than is otherwise found in introductory lectures on numerics. The book highlights the usefulness of matrix partitioning compared to a component view, leading not only to clearer notation and shorter algorithms but also to significant runtime gains in modern computer architectures. The algorithms and accompanying numerical examples are provided in the programming environment MATLAB, and additionally in an appendix in the future-oriented, freely accessible programming language Julia. This book is suitable for a two-hour lecture on numerical linear algebra from the second semester of a bachelor's degree in mathematics.