The book presents an introduction to the main topics addressed by Numerical Linear Algebra. At the same time, it describes the implementation of the required Scientific Computing, indicating the most suitable algorithm. Each proposed algorithm is analyzed in terms of complexity and stability and is explained with numerical examples. The book is the result of many years of teaching the course on Numerical Linear Algebra at the Department of Mathematics of the National and Kapodistrian University of Athens. The presentation of the material has been developed appropriately to serve as a study motivation for students in departments of Mathematics, Computer Science, Engineering Schools, Statistics, and other related departments. Particular emphasis has been placed on the following points: 1. Detailed presentation of floating-point arithmetic, 2. Development of error analysis theory, 3. Extensive presentation of matrix factorizations, 4. Least squares and Tikhonov regularization, 5. Introduction to large-scale computations and Krylov methods, 6. Presentation of a scientific computing lab. An innovation of this book is the presentation of selected scientific computations in the programming language JULIA. This language is particularly friendly for matrix computations and is freely available (open-source software). It is worth noting that in 2019, the prestigious “J. Wilkinson Prize for Numerical Software” was awarded to researchers who contributed to its development. The book is accompanied by the JULIA website, which provides readers with explanatory material and various useful information for the application of the language.