根据维基百科定义,kernel在线性代数和泛函分析中的定义为:
线性映射L : V ↦ W L:V\mapsto WL:V↦W,V和W为两个向量空间,满足L ( v ⃗ ) = 0 ⃗ L(\vec{v})=\vec{0}L(v)=0的所有元素v ⃗ \vec{v}v组成的空间,称为kernel或nullspace。
数学表示为:
k e r ( L ) = { v ⃗ ∈ V ∣ L ( v ⃗ ) = 0 } ker(L)=\{\vec{v}\in V|L(\vec{v})=0\}ker(L)={v∈V∣L(v)=0}
如上图所示,当两个不同的元素v 1 ⃗ , v 2 ⃗ \vec{v_1},\vec{v_2}v1,v2具有相同的image(W空间黄色区域内)时,则意味着v 1 ⃗ − v 2 ⃗ \vec{v_1}-\vec{v_2}v1−v2在L的kernel空间内:
L ( v 1 ⃗ ) = L ( v 2 ⃗ ) ⇔ L ( v 1 ⃗ − v 2 ⃗ ) = 0 ⃗ L(\vec{v_1})=L(\vec{v_2})\Leftrightarrow L(\vec{v_1}-\vec{v_2})=\vec{0}L(v1)=L(v2)⇔L(v1−v2)=0
看上图的黄色区域即左侧为源,右侧的黄色区域即为L的像。
左侧V源的Ker(L)的所有源都映射到右侧的0(向量)点。左侧V源除Ker(L)外的所有源点通过L都将映射到右侧的im(L)空间内,于是有:
i m ( L ) ≅ V / k e r ( L ) im(L)\cong V/ker(L)im(L)≅V/ker(L)
【In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by “collapsing” N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).】
根据rank-nullity定理
相应地有:dim(ker(L))+dim(im(L))=dim(V)。
举例如下:
参考资料:
[1] https://en.wikipedia.org/wiki/Kernel_(linear_algebra)
[2] https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem
[3] https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)