Textbook in PDF format Matrix decompositions are an important step in solving linear systems in a computationally efficient manner. Matrix decomposition methods, also called matrix factorization methods, are a foundation of linear algebra in computers, even for basic operations such as solving systems of linear equations, calculating the inverse, and calculating the determinant of a matrix. This is the age of Big Data. Every second of every day, data is being recorded in countless systems over the world. Our shopping habits, book and movie preferences, key words typed into our email messages, medical records, NSA recordings of our telephone calls, genomic data - and none of it is any use without analysis. Enormous data sets carry with them enormous challenges in data processing. Solving a system of 10 equations in 10 unknowns is easy, and one need not be terribly careful about methodology. But as the size of the system grows, algorithmic complexity and efficiency become critical. Matrix decompositions are an important step in solving linear systems in a computationally efficient manner. This book is comprised of seven chapters. Chapter 1 is about Vectors. Vectors may be visualized as directed line segments whose lengths are their magnitudes. Since only the magnitude and direction of a vector matter, any directed segment may be replaced by one of the same length and direction but beginning at another point, such as the origin of a coordinate system. Chapter 2 is about Matrices; Chapter 3: Linear algebra; Chapter 4: Matrices and Machines; Chapter 5: GaussElemination. Gaussian elimination is an algorithm for solving systems of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. Chapter 6: Rank Reducing Decomposition. Finally, in Chapter 7, the book focuses on the QR compositions and least squares