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数学分析笔记——An Introduction to Manifolds(3)

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:VW. 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 viR. Let αi:VR 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 ij.
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 vV 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:AA. 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(ar1)=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 τσ:AA, in that order: first apply σ, then τ.

A simple way to describe a permutation σ:AA is by its matrix
[12kσ(1)σ(2)σ(k)].

 

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