
数学分析笔记——An Introduction to Manifolds(3)
Chapter 1 Euclidean Spaces
§3 The Exterior Algebra of Multicovectors
3.1 Dual Space
If V and W are real vector spaces, we denote by Hom(V,W) the vector space of all linear maps f:V→W. Define the dual space V∨ of V to be the vector space of all real-valued linear functions on V:
V∨=Hom(V,R).
The elements of V∨ are called covectors or 1-covectors on V.
In the rest of this section, assume V to be a finite-dimensional vector space. Let e1,⋯,en be a basis for V. Then every v in V is uniquely a linear combination v=∑viei with vi∈R. Let αi:V→R be the linear function that picks out the
ith coordinate, αi(v)=vi. Note that αi is characterized by
αi(ej)=δij={1for i=j,0for i≠j.
Proposition. The functions α1,⋯,αn form a basis for V∨.
This basis α1,⋯,αn for V∨ is said to be dual to the basis e1,⋯,en for V.
Corollary. The dual space V∨ of a finite-dimensional vector space V has the same dimension as V.
Example (Coordinate functions). With respect to a basis e1,⋯,en for a vector space V, every v∈V can be written uniquely as a linear combination v=∑bi(v)ei, where bi(v)∈R. Let α1,⋯,αn be the basis of V∨ dual to e1,⋯,en. Then
αi(v)=αi(∑jbj(v)ej)=∑jbj(v)αi(ej)=∑jbj(v)δij=bi(v).
Thus, the dual basis to e1,⋯,en is precisely the set of coordinate functions b1,⋯,bn with respect to the basis e1,⋯,en.
3.2 Permutations
Fix a positive integer k. A permutation of the set A={1,⋯,k} is a bijection σ:A→A. More concretely, σ may be thought of as a reordering of the list 1,2,⋯,k from its natural increasing order to a new order σ(1),σ(2),⋯,σ(k).
The cyclic permutation, (a1 a2 ⋯ ar) where the ai are distinct, is the permutation σ such that σ(a1)=a2,σ(a2)=a3,⋯,σ(ar−1)=ar,σ(ar)=a1, and σ fixes all the other elements of A. A cyclic permutation (a1 a2 ⋯ ar) is also called a cycle of length r or an r-cycle.
A transposition is a 2-cycle, that is, a cycle of the form (a b) that interchanges a and b, leaving all other elements of A fixed.
Two cycles (a1 ⋯ ar) and (b1 ⋯ bs) are said to be disjoint if the sets {a1,⋯,ar} and {b1,⋯,bs} have no elements in common.
The product τσ of two permutations τ and σ of A is the composition τ∘σ:A→A, in that order: first apply σ, then τ.
A simple way to describe a permutation σ:A→A is by its matrix
[12⋯kσ(1)σ(2)⋯σ(k)].
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