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2011 | OriginalPaper | Buchkapitel

2. Vector Spaces

verfasst von : Nadir Jeevanjee

Erschienen in: An Introduction to Tensors and Group Theory for Physicists

Verlag: Birkhäuser Boston

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Abstract

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.

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Vorheriges Kapitel A Quick Introduction to Tensors

Nächstes Kapitel Tensors

Another word about axioms 3 and 4, for the mathematically inclined (feel free to skip this if you like): the axioms do not 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

https://static-content.springer.com/image/chp%3A10.1007%2F978-0-8176-4715-5_2/MediaObjects/148999_1_En_2_Equa_HTML.gif

and so the zero element is unique. Similarly, if −v and −v′ are both inverse to some vector v, then

https://static-content.springer.com/image/chp%3A10.1007%2F978-0-8176-4715-5_2/MediaObjects/148999_1_En_2_Equb_HTML.gif

and so inverses are unique as well.

Note also that any complex vector space is also a real vector space, since if you know how to multiply vectors by a complex number, then you certainly know how to multiply them by a real number. The converse, of course, is not true.

Hermitian matrices being those which satisfy A ^†≡(A ^T)^∗=A, where superscript T denotes the transpose and superscript ∗ denotes complex conjugation of the entries.

Another footnote for the mathematically inclined: as discussed later in this example, though, H _n(ℂ) is only a real vector space, so it is only a subspace of M _n(ℂ) when M _n(ℂ) is considered as a real vector space, cf. footnote 2.

The differential operator \(\Delta_{S^{2}}\) is also sometimes known as the spherical Laplacian, and is given explicitly by

https://static-content.springer.com/image/chp%3A10.1007%2F978-0-8176-4715-5_2/MediaObjects/148999_1_En_2_Equ4_HTML.gif

(2.4)

We will not 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 [14].

We do not 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.

See Hoffman and Kunze [10].

As mentioned in the preface, the ħ, which would normally appear in this expression, has been set to 1.

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.

This fact is proved in most real analysis books, see Rudin [13].

Throughout this text I will denote the identity operator or identity matrix; it will be clear from context which is meant.

Nomenclature to be justified in the next chapter.

We have again ignored the overall normalization of the spherical harmonics to avoid unnecessary clutter.

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.

In this case, (⋅∣⋅) is linear in the first argument as well as the second and would be referred to as bilinear.

These are often called ‘events’ in the physics literature.

We are, of course arbitrarily choosing the +++− signature; we could equally well choose −−−+.

See Rudin [13], for instance, for this and for proofs of all the claims made in this example.

The ⋅ in the notation (v|⋅) signifies the slot into which a vector w is to be inserted, yielding the number (v|w).

As long as we are 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 zeroth coordinate corresponds to time.

See Sakurai [14] or Gasiorowicz [5] or our discussion in Chap. 4.

S. Gasiorowicz, Quantum Physics, 2nd ed., Wiley, New York, 1996

10.

K. Hoffman and D. Kunze, Linear Algebra, 2nd ed., Prentice Hall, New York, 1971

13.

W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976

14.

J.J. Sakurai, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, Reading, 1994

16.

S. Sternberg, Group Theory and Physics, Princeton University Press, Princeton, 1994

Titel: Vector Spaces
verfasst von: Nadir Jeevanjee
Verlag: Birkhäuser Boston
Buch: An Introduction to Tensors and Group Theory for Physicists
Print ISBN: 978-0-8176-4714-8

Electronic ISBN: 978-0-8176-4715-5

Copyright-Jahr: 2011
DOI: https://doi.org/10.1007/978-0-8176-4715-5_2

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