This is Part I of this two-part book. The book is intended for undergraduate and graduate students of mathematics, engineering and social sciences in general. Professionals and researchers of those fields may also use it as a reference and find the book useful. It is one of the principal branches of mathematics. The contents are arranged in such a way that a beginner can easily grasp the material step by step. The theories are made lucid through illustrated examples. No prerequisite is necessary for sequential study of the materials, except some preliminary knowledge of calculus for the Calculus part of the book. The 'Generalized Inverse' is essential for undergraduate and graduate study in statistics, numerical analysis, engineering, physics and other social sciences of most of the institutions of the world.