Vector Spaces

Chapter 2 reviews the basic linear algebra essential for understanding tensors, and also develops some more advanced linear algebraic notions (such as dual spaces and non-degenerate Hermitian forms) which are also essential but usually are omitted in the standard ‘linear algebra for scientists and engineers’ course. This chapter also takes a more abstract point of view than that usually taken in lower-division linear algebra courses, in that it begins by discussing abstract vector spaces, which are defined axiomatically. This gives us the freedom to consider vector spaces made up of functions or matrices, rather than just vectors in Euclidean space. The utility of this is illustrated immediately through numerous physical examples. Following this, the elementary notions of span, linear independence, bases, components, and linear operators are discussed, and special care is taken to distinguish the component representation of vectors and linear operators from their existence as coordinate-free abstract objects. From here we move on to more advanced material, introducing dual spaces as well as non-degenerate Hermitian forms; the latter are the appropriate framework for the various scalar products that occur in physics. We conclude by showing how a non-degenerate Hermitian form allows us to turn vectors into dual vectors, which explains the relationship between bras and kets, as well as that between the covariant and contravariant components of a vector.