# 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 $\text{Hom}(V,W)$ the vector space of all linear maps $f:V\to W$. Define the dual space $V^\vee$ of $V$ to be the vector space of all real-valued linear functions on $V$:
$$V^\vee=\text{Hom}(V,\mathbb{R}).$$
The elements of $V^\vee$ are called covectors or $1$-covectors on $V$.

In the rest of this section, assume $V$ to be a finite-dimensional vector space. Let $e_1,\cdots,e_n$ be a basis for $V$. Then every $v$ in $V$ is uniquely a linear combination $v=\sum v^i e_i$ with $v^i\in\mathbb{R}$. Let $\alpha^i:V\to R$ be the linear function that picks out the
$i$th coordinate, $α^i(v)=v^i$. Note that $\alpha^i$ is characterized by
$$\alpha^i(e_j)=\delta_j^i=\begin{cases}1&\text{for }i=j,\\0&\text{for }i\not=j.\end{cases}$$
Proposition. The functions $\alpha^1,\cdots,\alpha^n$ form a basis for $V^\vee$.

This basis $\alpha^1,\cdots,\alpha^n$ for $V^\vee$ is said to be dual to the basis $e_1,\cdots,e_n$ for $V$.

Corollary. The dual space $V^\vee$ of a finite-dimensional vector space $V$ has the same dimension as $V$.

Example (Coordinate functions). With respect to a basis $e_1,\cdots,e_n$ for a vector space V, every $v\in V$ can be written uniquely as a linear combination $v=\sum b^i(v)e_i$, where $b^i(v)\in\mathbb{R}$. Let $\alpha^1,\cdots,\alpha^n$ be the basis of $V^\vee$ dual to $e_1,\cdots,e_n$. Then
$$\alpha^i(v)=\alpha^i\left(\sum\limits_jb^j(v)e_j\right)=\sum\limits_jb^j(v)\alpha^i(e_j)=\sum\limits_jb^j(v)\delta^i_j=b^i(v).$$
Thus, the dual basis to $e_1,\cdots,e_n$ is precisely the set of coordinate functions $b^1 ,\cdots,b^n$ with respect to the basis $e_1,\cdots,e_n$.

#### 3.2 Permutations

Fix a positive integer $k$. A permutation of the set $A=\{1,\cdots,k\}$ is a bijection $\sigma:A\to A$. More concretely, $\sigma$ may be thought of as a reordering of the list $1,2,\cdots,k$ from its natural increasing order to a new order $\sigma(1),\sigma(2),\cdots,\sigma(k)$.

The cyclic permutation, $(a_1\ a_2\ \cdots\ a_r)$ where the $a_i$ are distinct, is the permutation $\sigma$ such that $\sigma(a_1)=a_2,\sigma(a_2)=a_3,\cdots,\sigma(a_{r-1})=a_r,\sigma(a_r)=a_1$, and $\sigma$ fixes all the other elements of $A$. A cyclic permutation $(a_1\ a_2\ \cdots\ a_r)$ 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 $(a_1\ \cdots\ a_r)$ and $(b_1\ \cdots\ b_s)$ are said to be disjoint if the sets $\{a_1,\cdots,a_r\}$ and $\{b_1,\cdots,b_s\}$ have no elements in common.

The product $\tau\sigma$ of two permutations $\tau$ and $\sigma$ of $A$ is the composition $\tau\circ\sigma:A\to A$, in that order: first apply $\sigma$, then $\tau$.

A simple way to describe a permutation $\sigma:A\to A$ is by its matrix
$$\begin{bmatrix} 1&2&\cdots&k\\ \sigma(1)&\sigma(2)&\cdots&\sigma(k)\\ \end{bmatrix}.$$