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 solutioncommon in practical applicationsis 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 MoorePenrose 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.

Provide a practically applicable theory using only elementary linear algebra Deliver essential ideas for developing mathematics with real-world applications Introduce compelling practical applications across engineering and technology

Autorentext

Takashi Takiguchi is an associate professor at National Defense Academy of Japan.

Inhalt

Introduction; motivation of this monograph.- General theory of least square solutions to an overdetermined system of linear equations with no solution.- Semi-equivalent systems to overdetermined systems of linear equations with no solution.- Practical applications.- Similar ideas to analyze an overdetermined system of linear equations with no solution.