Linear Algebra and Analytical Geometry

This course is intended for students who have already basic math knowledge and need to strengthen their learning on higher level mathematics. It covers two areas, Linear Algebra and Analytical Geometry. Each part of the course contains: definition of mathematical concepts, new mathematical terms, many useful examples, resolutions of exercises, proposed exercises and their solutions.

Linear Algebra is one of the most important areas in Mathematics. Its impact is, at least, as great as Calculus, and indeed it provides a significant part of the machinery required to generalise Calculus to vector-valued functions of many variables. Many difficult science problems can be handled using the powerful, yet easy to use, mathematics of linear algebra. Many geometric topics are studied by using concepts from linear algebra, and the idea of linear transformations, is an algebraic version of geometric transformation.
Analytical Geometry basically deals with the same geometric objects as the elementary geometry does. The difference is in the method for studying these objects. The elementary geometry relies on visual impressions and formulates the properties of geometric objects in its axioms. From these axioms, various theorems are derived, whose statements in most cases are also revealed in visual impressions. The analytical geometry is more inclined to a numeric description of geometric objects and their properties. This chapter on analytic geometry is in two and three dimensions. Two by two matrices are interpreted as geometric transformations and it is shown how the arithmetic of matrices can be used to obtain significant geometrical results.

Enjoy the reading!