# 向量空间和子空间

## 1. 向量空间 (vector space)

a vector space is a nonempty set $$\mathcal{V}$$ of object (elementary), called vectors, on which are defined two operations, called addition and multiplication by scalars (real number or complex number), subject to the ten axioms below. The axioms must hold for all $$\mathbf u$$ , $$\mathbf v$$ and $$\mathbf w$$ in $$\mathcal{V}$$ and for all scalars $$c$$ and $$d$$.

1. $$\mathbf {u+v}$$ is in $$\mathcal{V}$$.
2. $$\mathbf {u+v = v + u}$$ .
3. $$\mathbf {(u+v) + w = u + (v +w)}$$
4. There is a vector (called the zero vector) $$\mathbf 0$$ in $$\mathcal{V}$$ such that $$\mathbf {u+ 0 = u}$$
5. For each $$\mathbf u$$ in $$\mathcal{V}$$, there is vector $$\mathbf {-u}$$ in $$\mathcal{V}$$ satisfying $$\mathbf {u+(-u)=0}$$
6. $$c \mathbf u$$ is in $$\mathcal{V}$$.
7. $$c \mathbf {(u+v)} = c \mathbf u + c \mathbf v$$
8. $$(c+d) \mathbf u = c \mathbf u + d \mathbf u$$
9. $$(cd) \mathbf u = c (d \mathbf u)$$
10. $$1\mathbf u = \mathbf u$$

:

• 向量空间对加法 (addition) 和数乘 (scalar multiplication ) 运算封闭
• 向量空间 (vector space) 也称线性空间 (linear space)

${\mathcal {P}}_{n}(t) = a_0 + a_1 t + a_2 t^2 + \cdots + a_n t^n$

### 1.1 常用的向量空间 (线性空间)

$$\mathbb C^N$$ : Vector space of complex-valued finite-dimension vectors $\mathbb C^N = \big \lbrace \mathbf {x} = [x_0 \;\; x_1 \;\cdots \; x_{N-1}]^T \big | x_n \in \mathbb C, n \in \lbrace0, 1, \ldots, N-1 \rbrace \big \rbrace$

$$\mathbb C^\mathbb Z$$ : Vector space of complex-valued sequence over $$\mathbb Z$$ $\mathbb C^\mathbb Z = \big \lbrace \mathbf {x} = [\cdots \; x_1 \; \; x_0 \;\; x_1 \;\cdots]^T \big | x_n \in \mathbb C, n \in \mathbb Z \big \rbrace$
$$\mathbb C^\mathbb R​$$ : Vector space of complex-valued sequence over $$\mathbb R​$$ $\mathbb C^\mathbb R= \big \lbrace \mathbf x \; \big | \; \mathbf x(t) \in \mathbb C, t \in \mathbb R \big \rbrace$

## 2. 线性子空间 (linear subspace)

A linear subspace of a vector space (linear space) $$\mathcal{V}$$ is a subset $$\mathcal{H}$$ of $$\mathcal{V}$$ that has three properties:

• The zero vector $$\{ \mathbf 0\} = \mathcal {O}$$ of $$\mathcal{V}$$ is in $$\mathcal{H}$$
• For each $$\mathbf u$$ and $$\mathbf u$$ are in $$\mathcal{H}$$ , $$\mathbf {u+v}$$ is in $$\mathcal{H}$$ (In this case we say $$\mathcal{H}$$ is closed under vector addition)
• For each $$\mathbf u$$ in $$\mathcal{H}$$ and each scalar $$c$$ , $$c \mathbf u$$ is in $$\mathcal{H}$$ . (In this case we say $$\mathcal{H}$$ is closed under scalar multiplication)

1. $$\mathcal {H} + \mathcal {H} = \mathcal {H}$$
2. $$\lambda \mathcal {H} = \mathcal {H}$$ ，其中 $$\lambda \in \mathbb F$$

1. 如果 $$\mathcal {X} = \mathbb R^2$$, 那么所有过原点的直线都是不平凡的 (nontrivial) 线性子空间
2. $$\mathcal {H} = \lbrace \mathbf{x} = (0, x_2, x_3, \ldots , x_n) : x_i \in \mathbb R, i = 2, 3, \ldots, n \rbrace$$$$\mathbb R^n$$ 的子空间
3. $$\mathcal {H} = \lbrace x: [-1, 1] \to \mathbb R, x \; \text {continuous and} \; x(0) = 0 \rbrace$$ is a subspace of $$\mathcal {C}[-1, 1]$$
4. $$\mathcal {H} = \lbrace x: [-1, 1] \to \mathbb R, x \; \text {continuous and} \; x(0) = 1 \rbrace$$ is not a subspace of $$\mathcal {C}[-1, 1]$$

### 2.1 线性独立 (linear independent) 和基 (basis)

• 基之间线性独立
• 基的线性组合填满整个线性空间 $$\mathcal {X}$$ ，或者说 $$\{ \mathbf {v}_1, \mathbf {v}_2, \ldots, \mathbf {v}_n \}$$ 生成(span) $$\mathcal {X}$$， 又或者说 $$\mathcal {X}$$ 的每一个向量都可以表示为 $$\{ \mathbf {v}_1, \mathbf {v}_2, \ldots, \mathbf {v}_n \}$$ 的线性组合

1. 空间 $$\mathbb R^n$$ 的维度为 $$n$$ , 它的标准基为 $$\{ \mathbf {e}_1, \mathbf {e}_2, \ldots, \mathbf {e}_n \}$$ ，其中对于 $$j = 1, 2, …, n$$， 我们有 $$\mathbf {e}_j = (0, 0, …, 1, 0, …, 0)$$ ，只有第 $$j$$ 个元素为 $$1$$
2. 前面提到的空间 $${\mathcal {P}}_{n}$$ – 最多 $$n$$ 阶的多项式函数空间，它的维度为 $$n+1$$ ，它的标准基底为 $$\{ 1, t, t^2, \ldots, t^n \}$$
3. 函数空间 $$\mathcal {C}[a, b]$$ 是无限维的 (infinite-dimensional)

### 2.2 仿射空间 affine space

If $$\mathcal {C}$$ is an affine set and $$\mathbf {v}_0 \in \mathcal {C}$$, then the set $$\mathcal {V}$$ is a subspace (子空间). $\mathcal {V} = \mathcal {C} – \mathbf {v}_0 = \{ \mathbf {x} – \mathbf {v}_0 | \mathbf {x} \in \mathcal {C} \}$

$$\mathrm {aff} \{\mathbf {v}_1, \mathbf {v}_2 , \mathbf {v}_3 \}$$ 就是 $$\mathrm {span} \{ \mathbf {v}_2 -\mathbf {v}_1, \mathbf {v}_3 -\mathbf {v}_1 \}$$ 平移 $$\mathbf {v}_1$$ 而得的平面，也就是过 $$\mathbf {v}_1, \mathbf {v}_2, \mathbf {v}_3$$ 三点的平面，见下图:

### 2.3 凸集

• $$\mathcal {K}$$ 是凸集(convex) 如果对于任何 $$\mathbf {x}, \mathbf {y} \in \mathcal {K}$$ 并且 $$\lambda \in [0, 1]$$ ，有 $$\lambda \mathbf {x} + (1-\lambda) \mathbf {y} \in \mathcal {K}$$
• $$\mathcal {K}$$ is balanced ，如果对于任何 $$\mathbf {x} \in \mathcal {K}$$$$|\lambda| \le 1$$ ，有 
• $$\mathcal {K}$$ is absolutely convex 如果 $$\mathcal {K}$$ 是 convex and balanced

1. $$\mathcal {K}$$ is absolutely convex if and only if $$\lambda \mathbf {x} + \mu \mathbf {y} \in \mathcal {K}$$ whenever $$\mathbf {x}, \mathbf {y} \in \mathcal {K}$$ and $$|\lambda| + |\mu| \le 1$$
2. 任何线性空间都是 absolutely convex

$$co(\mathcal {S})$$ 是包含集合 $$\mathcal {S}$$ 的最小凸集，表示为: $co(\mathcal {S}) = \Big \lbrace {\sum_{j=1}^{n}} \lambda_j {{\mathbf {x}}_j} \vert {{\mathbf {x}}_1}, {{\mathbf {x}}_2}, \ldots, {{\mathbf {x}}_n} \in \mathcal {S}, \lambda_j \ge 0, \forall \; j=1,2, \ldots n, {\sum_{j=1}^{n}} \lambda_j=1, n \in \mathbb N \Big \rbrace$