图5 Vector addition and scalar multiplication (source from here)
其他例子,还包括坐标空间(Coordinate spaces)、复数、函数空间(Function spaces)、线性方程组(linear equations)。详情可查阅维基百科词条:Examples of vector spaces.
5.1 8个公理
摘抄维基百科Vector space部分内容如下:
A vector space is a collection of objects called vectors, which may be added together and multiplied(“scaled”) by numbers, called scalars in this context. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.
给定域F,向量空间V记为F-向量空间。其二元运算:
向量加法:+ : V × V → V 记作 v + w, ∃ v, w ∈ V
标量乘法:·: F × V → V 记作 a v, ∃a ∈ F 且 v ∈ V
并且满足如下8条公理[10]:
向量加法结合律:u + (v + w) = (u + v) + w
向量加法的单位元:V存在零向量的0,∀ v ∈ V , v + 0 = v
向量加法的逆元素:∀v∈V, ∃w∈V,使得 v + w = 0
向量加法交换律:v + w = w + v
标量乘法与域乘法兼容性(compatibility): a(b v) = (ab)v
标量乘法有单位元: 1 v = v, 1指域F的乘法单位元
标量乘法对于向量加法满足分配律:a(v + w) = a v + a w
标量乘法对于域加法满足分配律: (a + b)v = a v + b v
另,若F是实数域ℝ,则V称为实数向量空间;若F是复数域ℂ,则V称为复数向量空间;若F是有限域,则V称为有限域向量空间****
图5 Vector addition and scalar multiplication (source from here)
其他例子,还包括坐标空间(Coordinate spaces)、复数、函数空间(Function spaces)、线性方程组(linear equations)。详情可查阅维基百科词条:Examples of vector spaces.
5.1 8个公理 摘抄维基百科Vector space部分内容如下:
A vector space is a collection of objects called vectors, which may be added together and multiplied(“scaled”) by numbers, called scalars in this context. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.
给定域F,向量空间V记为F-向量空间。其二元运算:
向量加法:+ : V × V → V 记作 v + w, ∃ v, w ∈ V 标量乘法:·: F × V → V 记作 a v, ∃a ∈ F 且 v ∈ V 并且满足如下8条公理[10]:
向量加法结合律:u + (v + w) = (u + v) + w 向量加法的单位元:V存在零向量的0,∀ v ∈ V , v + 0 = v 向量加法的逆元素:∀v∈V, ∃w∈V,使得 v + w = 0 向量加法交换律:v + w = w + v 标量乘法与域乘法兼容性(compatibility): a(b v) = (ab)v 标量乘法有单位元: 1 v = v, 1指域F的乘法单位元 标量乘法对于向量加法满足分配律:a(v + w) = a v + a w 标量乘法对于域加法满足分配律: (a + b)v = a v + b v 另,若F是实数域ℝ,则V称为实数向量空间;若F是复数域ℂ,则V称为复数向量空间;若F是有限域,则V称为有限域向量空间****