The Least Square Solution to an Overdetermined System

This book addresses a gap in elementary linear algebra: while solving systems of linear equations with multiple solutions is standard, analyzing overdetermined systems with no solution-common in practical applications-is rarely taught. We develop a theory for such systems using the concept of least-squares solutions, grounded entirely in elementary linear algebra. Unlike approaches based on the Moore-Penrose inverse, which require advanced tools such as singular value decomposition, our method relies only on basic principles yet performs effectively in real-world contexts. Practical applications include non-destructive inspection of concrete structures, GNSS-based position detection, and the implementation of computerized tomography. Designed as a monograph for researchers and educators, this book can also serve as a graduate or advanced undergraduate textbook for students interested in applied mathematics. It is equally suitable for senior undergraduate research projects and for engineers seeking accessible mathematical tools for practical problems.