Vector Spaces

Nadir Jeevanjee
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Nadir Jeevanjee: University of California at Berkeley, Department of Physics

Chapter Chapter 2 in An Introduction to Tensors and Group Theory for Physicists, 2015, pp 11-50 from Springer

Abstract: Abstract This chapter reviews the basic linear algebra essential for understanding tensors (linear independence, bases, linear operators, etc.), and also develops some more advanced linear algebraic notions (e.g., dual spaces and non-degenerate Hermitian forms) which are also essential but often undiscussed. This chapter also takes a more abstract point of view than is typical, which gives us the freedom to consider vector spaces made up of functions or matrices, rather than just vectors in Euclidean space. Throughout, special care is taken to distinguish the component representation of various objects (vectors, linear operators, etc.) from their existence as coordinate-free abstract objects. The machinery developed is also used to illuminate enigmatic topics such as spherical harmonics and the relationship between bras and kets and the covariant and contravariant components of a vector.

Date: 2015
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DOI: 10.1007/978-3-319-14794-9_2

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