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.
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Notes
- 1.
Meaning that these operations always produce another member of the set V, i.e. a vector.
- 2.
Another word about axioms 3 and 4, for the mathematically inclined (feel free to skip this if you like): the axioms don’t demand that the zero element and inverses are unique, but this actually follows easily from the axioms. If 0 and 0′ are two zero elements, then
$$\displaystyle{0 = 0 + 0' = 0',}$$and so the zero element is unique. Similarly, if − v and − v′ are both inverse to some vector v, then
$$\displaystyle{-v' = -v' + 0 = -v' + (v - v) = (-v' + v) - v = 0 - v = -v,}$$and so inverses are unique as well.
- 3.
Hermitian matrices being those which satisfy \(A^{\dag }\equiv (A^{T})^{{\ast}} = A\) where superscript T denotes the transpose and superscript * denotes complex conjugation of the entries.
- 4.
Another footnote for the mathematically inclined: as discussed later in this example, though, \(H_{n}(\mathbb{C})\) is only a real vector space, so it is only a subspace of \(M_{n}(\mathbb{C})\) when \(M_{n}(\mathbb{C})\) is considered as a real vector space.
- 5.
The differential operator \(\Delta _{S^{2}}\) is also sometimes known as the spherical Laplacian, and is given explicitly by
$$\displaystyle{ \Delta _{S^{2}} = \frac{\partial ^{2}} {\partial \theta ^{2}} +\cot \theta \frac{\partial } {\partial \theta } + \frac{1} {\sin ^{2}\theta } \frac{\partial ^{2}} {\partial \phi ^{2}}. }$$(2.6)We won’t need the explicit form of \(\Delta _{S^{2}}\) here. A derivation and further discussion can be found in any electrodynamics or quantum mechanics book, like Sakurai [17].
- 6.
We don’t generally consider infinite linear combinations like \(\sum _{i=1}^{\infty }c^{i}v_{ i} =\lim _{N\rightarrow \infty }\sum _{i=1}^{N}c^{i}v_{ i}\) because in that case we would need to consider whether the limit exists, i.e. whether the sum converges in some sense. More on this later.
- 7.
See Hoffman and Kunze [13].
- 8.
If you take \(C = \mathbb{R}\), then you need to multiply the basis vectors in (2.10) by i and add them to the basis set, giving \(\mathbb{C}^{n}\) a real dimension of 2n.
- 9.
As mentioned in the preface, the \(\hslash \) which would normally appear in this expression has been set to 1.
- 10.
The simple form of \([e'_{1}]_{\mathcal{B}'}\) is no accident; you can easily check that if you express any set of basis vectors in the basis that they define, the resulting column vectors will just look like the standard basis.
- 11.
This fact is proved in most real analysis books; see Rudin [16].
- 12.
Throughout this text I will denote the identity operator or identity matrix; it will be clear from context which is meant.
- 13.
Nomenclature to be justified in the next chapter.
- 14.
We have again ignored the overall normalization of the spherical harmonics to avoid unnecessary clutter.
- 15.
If V is infinite-dimensional, then this may not work as the sum required may be infinite, and as mentioned before care must be taken in defining infinite linear combinations.
- 16.
In this case, (⋅ | ⋅ ) is linear in the first argument as well as the second and would be referred to as bilinear.
- 17.
These are often called “events” in the physics literature.
- 18.
We are, of course, arbitrarily choosing the \(+ + +-\) signature; we could equally well choose \(---+\).
- 19.
See Rudin [16], for instance, for this and for proofs of all the claims made in this example.
- 20.
The ⋅ in the notation (v | ⋅ ) signifies the slot into which a vector w is to be inserted, yielding the number (v | w).
- 21.
As long as we’re talking about “standard” physics notation, you should also be aware that in many texts the indices run from 0 to 3 instead of 1 to 4, and in that case the 0th coordinate corresponds to time.
- 22.
We could use non-degenerate Hermitian forms here rather than inner products to make a similar point, but will stick with inner products for definiteness.
- 23.
References
S. Gasiorowicz, Quantum Physics, 2nd edn. (Wiley, New York, 1996)
K. Hoffman, D. Kunze, Linear Algebra, 2nd edn. (Prentice Hall, Englewood Cliffs, 1971)
W. Rudin, Principles of Mathematical Analysis, 3rd edn. (McGraw Hill, New York, 1976)
J.J. Sakurai, Modern Quantum Mechanics, 2nd edn. (Addison Wesley Longman, Reading, MA, 1994)
S. Sternberg, Group Theory and Physics (Princeton University Press, Princeton, 1994)
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Jeevanjee, N. (2015). Vector Spaces. In: An Introduction to Tensors and Group Theory for Physicists. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-14794-9_2
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