Linear algebra and geometry (Q2785497)

This book originates from lecture notes. It begins with motivational or review material dealing with vectors and mappings in the plane, thus establishing some basic knowledge. Sets and their properties, including Zorn's lemma come first. Elementary properties of groups, rings, and fields lay the groundwork for a rigorous study of linear algebra. NEWLINENEWLINENEWLINEThe theory of vector spaces over arbitrary fields, of linear transformations and of matrices is presented. This includes normal forms of matrices, in particular the Jordan normal form. Solving of systems of linear equations is taught, followed by a taste of numerical considerations and error analysis. NEWLINENEWLINENEWLINEThe chapter on Euclidean and unitary vector spaces contains normal forms of orthogonal and unitary transformations as well as an outlook on Banach spaces, Banach algebras, and ordinary differential equations. There are chapters on linear programming and on multilinear algebra. Normal forms of Abelian groups and the Sylow theorems are proved to round off group theory. A detailed study of analytic affine and projective geometry includes conic sections and quadratic forms. NEWLINENEWLINENEWLINEA discussion of the Erlangen Program is followed by a study of the Poincaré model and elliptic geometry. A taste of the foundations of geometry is provided in the form of Hilbert's system of axioms. Numerous theorems for the Euclidean plane are stated, some are proved. In the last section of this book the student gets some guidance in programming algorithms for the computer.