Textbook in PDF format In mathematics, most areas deal with various types of spaces and study functions defined between these spaces, as well as how these functions preserve or do not preserve various properties of these spaces. Linear algebra, in particular, deals with spaces in which objects can be scaled by a scalar and added together, and studies functions that preserve such properties. We refer to these objects, spaces, and functions as vectors, vector spaces, and linear transformations, respectively. We use linearity to collectively refer to the scalability and additivity of vectors that are preserved under linear transformation across vector spaces. This linearity enables us to understand the universal structures of vector spaces and linear transformations. For example, vector spaces of the same dimension are roughly equivalent. We also found that a linear transformation can be characterized by a rectangular array of numbers called a matrix. Then, to investigate and classify linear transformations, we work with matrices corresponding to linear transformations, and this allows us to successfully classify these linear transformations into Jordan forms. Such fundamental efforts for classification have led to a variety of by-products that have proven to be useful for many real-world applications. In this book, we do not make a compromise between the fundamental results and useful by-products, by providing readers with gap-free derivations of useful by-products from the fundamental results