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Definition (Kernel)
Given the linear map T∈L(V,W)T\in\mathscr{L}(V,W)T∈L(V,W), we define the kernel of TTT to be Ker(T):={v∈V∣T(v)=0}\text{Ker}(T):=\{v\in V|T(v)=0\}Ker(T):={v∈V∣T(v)=0}
Remark
We can think of the kernel as the set of elements in the input set that map to the 0 vector.
Proposition (Injective ⟺ Ker(T)=0\iff Ker(T)=0⟺Ker(T)=0)
A Linear Map T∈L(V,W)T\in\mathscr{L}(V,W)T∈L(V,W) is Injective if and only if Ker(T)=0Ker(T)=0Ker(T)=0.
Eigenvector
Kernel
Rank Nullity Theorem