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An Introductory Course on Differentiable Manifolds
By Siavash Shahshahani Dover Publications, Inc.
Copyright © 2016 Siavash Shahshahani
All rights reserved.
ISBN: 978-0-486-82082-8
CHAPTER 1
Multilinear Algebra
In this chapter, we discuss various structures and mappings that involve one or several vector spaces over a fixed field. It will always be assumed that the field characteristic is zero; in fact, the reader may assume that the underlying field is R or C. In Section D, we will consider special features of vector spaces over R.
A. Dual Space
Let V be a finite dimensional vector space over a field F. The set of linear mappings V ->F will be denoted by V*. This set will be endowed with the structure of a vector space over F. Let α and β be elements of V* and r an element of F, then we define α+β and rα by
(1.1) (α+β)(x) = α(x) + β(x)
(1.2) (rα)(x) = rα(x)
where x is an arbitrary element of V. With these operations, V* becomes a vector space over F and will be called the dual space to V. Suppose (e1, ..., en) is a basis for V. We define elements ei of V* by their value on ej, j = 1, ..., n, as follows:
(1.3) ei(ej) = σij
where σi denotes the value 1 or 0 depending on whether i=j or i ≠ j. Note that any element α of V* can be written as a linear combination of e1, ..., en. In fact,
[MATHEMATICAL EXPRESSION OMITTED]
since the value of both sides on an arbitrary basis element ej is the same. Further, {e1, ..., en} is a linearly independent set, for if [summation]iriei=0, evaluating both sides on the basis element ej yields rj=0. Therefore, the ordered set (e1, ..., en) is a basis for V*, called the dual basis for V* relative to (e1, ..., en). Thus V* has the same dimension as V.
By repeating the operation of dual making, one can look at (V*)*, the so-called double dual of V, usually denoted by V**. The double dual will then have the same dimension as the original space V, and since all linear spaces of the same dimension over a given field are isomorphic, there are isomorphisms between V, V* and V**. But in the case of V and V**, there is a distinguished natural isomorphism, denoted by IV:V ->V**, which is given as follows. For each v [member of] V, the element IV(v) is defined by
(1.4) (IV(v))(α) = α(v)
It follows from (1.1) and (1.2) that IV(v) is indeed linear, i.e., it is a member of V**. That IV is linear follows from the linearity of α. To show that IV is an isomorphism, it suffices to show that its kernel is {0} since the domain and target linear spaces are finite dimensional of the same dimension. But α(v)=0 for all α in V* implies that v=0, and the isomorphism is established. Note that the definition of IV was independent of the specific nature of the linear space V or the choice of basis for it. In fact, one can state the following general assertion.
1. TheoremFor any basis (e1, ..., en) of V, (IV (e1), ..., IV(en])ITL) is the dual basis in V** relative to the basis (e1, ..., en) for V*.
Proof. We must show
[MATHEMATICAL EXPRESSION OMITTED]
This is a consequence of (1.4) and (1.3).
By virtue of the natural isomorphism IV, the space V** is often identified with V. Under this identification, IV(ei) is identified with ei, so that (e1, ..., en) becomes the dual basis for V** relative to (e1, ..., en).
B. Tensors
Let V1, ..., Vp and W be vector spaces over a field. A map α: V1 X ... X Vp ->W is called p-linear provided that by fixing any p-1 components of (v1, ..., vp) [member of] V1 X ... X Vp, α is linear with respect to the remaining component. As we shall see in some of the following examples, operations generally known as "products" in elementary mathematics are of this nature.
2. Examples
(a) Let V be a vector space over a field F. Regard F as a one-dimensional vector space over F. Then the product F X V ->V given by
(r; v) [??] rv
is 2-linear (bilinear).
(b) Let F be a field. Then the p-fold product F X ... X F ->F given by
(r1, ..., rp) [??] r1 ... rp
is p-linear.
(c) Let V be a vector space over R. Then any inner product V X V -> R is bilinear. The vector product R3 X R3 -> R3 is another example of a bilinear mapping. In general, let β: V X V ->F be bilinear and consider a basis (e1, ..., en) for V. The nXn matrix B=[βij], where βij = β(ei; ej), determines β completely as
(1.5) [MATHEMATICAL EXPRESSION OMITTED]
If B is a symmetric matrix with positive eigenvalues, then β is an inner product.
Conversely, any inner product on V is obtained in this manner.
(d) For a vector space V over a field F, the evaluation pairing V*V* ->F, given by (v, α) [??]α (v), is bilinear.
(e) Let F be a field and V1, ..., Vp; W1, ..., Wq be vector spaces over F. Suppose p-linear and q-linear maps α:V1 X ... Vp ->F and β:W1 X ... X Wq -> F are given. Then the tensor product
[MATHEMATICAL EXPRESSION OMITTED]
is defined by
(1.6) [MATHEMATICAL EXPRESSION OMITTED]
Note that α [cross product] β is a (p+q)-linear mapping. Further, it follows from the associativity of the product operation in the field F that is associative, hence the product α1 [cross product] ... [cross product] αk is unambiguously defined by induction.
In what follows, V will be a finite dimensional vector space over a field F. The n-fold product V X ... X V will be denoted by Vn.
3. Definition
(a) A p-linear map Vp ->F will be called a covariant p-tensor, or a tensor of type (p,0), on V.
(b) A q-linear map (V*)q ->F will be called a contravariant q-tensor, or a tensor of type (0,q), on V.
(c) A (p + q)-linear map Vp(V*)q ->F will be called a mixed (p,q)-tensor, or a tensor of type (p,q), on V.
4. Examples An element of V* is a covariant 1-tensor on V. In view of the natural isomorphism IV, any member of V may be regarded as a contravariant tensor on V. The evaluation pairing (Example 2d) is a (1; 1)-tensor on V. Inner products are examples of covariant 2-tensors.
We use the symbols Lp(V), Lq(V) and Lpq (V), respectively, to denote the sets of (p; 0)-, (0; q)- and (p; q)-tensors on V. Under functional addition, and multiplication by elements of the field F, each of these becomes a vector space over F. The dimensions of these spaces are, respectively, np, nq and np+q, as the following will imply.
5. Basis for the Space of TensorsLet (e1, ..., en) be a basis for V. Then the following are basis elements for the spaces of tensors.
(a) For Lp(V):
(1.7) [MATHEMATICAL EXPRESSION OMITTED]
(b) For Lq(V):
(1.8) [MATHEMATICAL EXPRESSION OMITTED]
(c) For Lpq (V):
(1.9) [MATHEMATICAL EXPRESSION OMITTED]
Proof. Note that by virtue of Example 2e, the displayed tensors are actually elements of the stated spaces. We prove the third case which includes the other two. To show linear independence, suppose that
[MATHEMATICAL EXPRESSION OMITTED]
By applying the two sides to ([MATHEMATICAL EXPRESSION OMITTED]), we see that the coefficients are zero, and linear independence is established. On the other hand, note that any α [member of] Lpq can be written as
(1.10) [MATHEMATICAL EXPRESSION OMITTED]
which can be verified by applying both sides to ([MATHEMATICAL EXPRESSION OMITTED]).
By convention, we let L0V=L0V=F.
6. Change of Basis
The bases introduced above for the spaces of tensors as well as the resulting components of the tensors depend on the original choice of basis for the linear space.
We are now going to investigate how a linear change of basis for the space affects the value of tensor components. We take V to be an n-dimensional vector space over F. It will be convenient to write nXn matrices with entries from F as A=[aij], where the superscript denotes the row index and the subscript indicates the column of the matrix entry. Suppose two bases B=(e1, ..., en) and [bar.B] = ([bar.e]1, ..., [bar.e]n) are given for V, related linearly by matrix A=[aij] as
(1.11) [MATHEMATICAL EXPRESSION OMITTED]
Thus the components of [bar.e]j with respect to the basis B are the entries of the jth column of matrix A. Corresponding to B and [bar.B], we have the dual bases B*=(e1, ..., en) and [bar.B]*=([bar.e]1, ..., [bar.e]n). We will first investigate the linear relationship between these two bases. We write
(1.12) [MATHEMATICAL EXPRESSION OMITTED]
Therefore, the components of [bar.e]i with respect to the basis B* are the entries of the ith row of matrix B=[bij]. To identify B, we note that
[MATHEMATICAL EXPRESSION OMITTED]
Therefore, the matrix B is the inverse of the transpose of the matrix A:
B-1 = AT
Now let α be a (p, q)-tensor on V. With respect to the above bases, the following two representations for α are obtained.
[MATHEMATICAL EXPRESSION OMITTED]
We wish to express the components [MATHEMATICAL EXPRESSION OMITTED] in terms of [MATHEMATICAL EXPRESSION OMITTED]. Using (1.10), we have
[MATHEMATICAL EXPRESSION OMITTED]
This is equal to
[MATHEMATICAL EXPRESSION OMITTED]
Thus we have obtained the desired formula for the change of tensor components under a linear change of variables
(1.13) [MATHEMATICAL EXPRESSION OMITTED]
Classically, a tensor is defined as a collection of numerical quantities [MATHEMATICAL EXPRESSION OMITTED] which transform under a linear change of variables according to formula (1.13).
(a) Special case (p=1,q=0) For a covariant 1-tensor
[MATHEMATICAL EXPRESSION OMITTED]
we obtain
(1.14) [MATHEMATICAL EXPRESSION OMITTED]
(b) Special case (p=0,q=1) Consider a contravariant 1-tensor, or by virtue of the natural isomorphism IV, an element x of V
[MATHEMATICAL EXPRESSION OMITTED]
In this case, we have
(1.15) [MATHEMATICAL EXPRESSION OMITTED]
7. Functoriality
Let V and W be vector spaces over a field F, and suppose f:V ->W is a linear map. For each non-negative integer p, a map Lpf:LpW ->LpV is defined as follows. If p = 0, L0f=1F. For p > 0, suppose α [member of] LpW and v1, ..., vp [member of] V, then
(1.16) [MATHEMATICAL EXPRESSION OMITTED]
That (Lp f)(α) [member of] LpV follows from the linearity of f and the fact that α [member of] LpW. The linearity of Lp f follows from the definition of linear space operations in the space of tensors. The following two properties are straightforward consequences of definition and establish Lp as a contravariant functor.
(a) For any vector space V and any non-negative integer p,
(1.17) [MATHEMATICAL EXPRESSION OMITTED]
(b) For linear maps f : V ->W and g : U ->V, and any non-negative integer p,
(1.18) Lp(f ο g) = Lpg ο Lp f
Of course, L1V = V*. The induced linear map L1f is denoted by f*. Note that by definition, LqV* = LqV. For a linear map f : V ->W, we denote Lqf* by Lqf. The following properties follow from (a) and (b) above and are summarized by saying that Lq is a covariant functor.
(c) For any vector space V and non-negative integer q,
(1.19) [MATHEMATICAL EXPRESSION OMITTED]
(d) For linear maps f : V ->W and g : U ->V, and any non-negative integer q,
(1.20) Lq(f ο g) = Lq f ο Lqg
C. Anti-symmetric Tensors
The so-called anti-symmetric tensors are among the most powerful tools in the study of geometric structures. As we shall see in the following section, these are closely related to the concepts of volume and orientation in the case of real vector spaces.
We recall some elementary facts about the group Sn of permutations on n symbols {1, ..., n}. A transposition is a permutation that exchanges two symbols and leaves the other symbols fixed. Any permutation σ [member of] Sn can be written as a composition of transpositions, σ_=_τ1 ο ... οτk, where k is not unique but its parity (even- or oddness) is determined by σ. Thus a permutation σ is called even or odd depending on whether k is even or odd. We write ε(σ) = +1 or ε(σ) = -1, respectively, if σ is even or odd. The map ε, called the sign, is a homomorphism from Sn onto the two-element multiplicative group {+1; -1}; thus ε(σ1 ο σ2)=ε(σ1) ο ε(σ2) and [MATHEMATICAL EXPRESSION OMITTED]. The set of even permutations form a subgroup of index 2 in Sn.
(Continues...)
Excerpted from An Introductory Course on Differentiable Manifolds by Siavash Shahshahani. Copyright © 2016 Siavash Shahshahani. Excerpted by permission of Dover Publications, Inc..
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