The Role of Bases

In this chapter, we discuss in detail the basics of Linear Algebra. The first important concepts in a vector space are linear combinations of vectors and related notions like generating systems, linear independent and linear dependent systems. This leads directly to the central notions of bases of vector spaces and their dimension. Bases allow us to perform concrete calculations for vectors needed in physics, such as assigning a list of numbers, the coordinates. This enormous advantage has its price. The representation of an abstract vector by a list of numbers depends on the chosen basis. Any theoretical calculation we do should obviously not depend on the choice of a basis. We discuss in detail the satisfactory but demanding solution to this problem in Sect. 3.2 which can be skipped on a first reading. We then demonstrate a suitable choice of basis for the representation of linear maps, and we discuss the origin of tensors in an elementary manner. Finally, we provide an important application for physics, and show that the transition from Newtonian mechanics to Lagrangian mechanics is nothing but the transition from a linear dependent to a linear independent system.