In Finite Dimensions Injectivity and Surjectivity are Equivalent Theorem

Theorem

• Suppose $T:V\to W$
• Suppose $dim(V)=dim(W)=n<\infty$ (T is linear)
• T is Injective $⟺$ T is Surjective

All or Nothing Matrix

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Text Elements

Injective

Yes

No

No

Yes

Surjective

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Proof

1. Rank Nullity Theorem says $dim(V)=dim(Image(T))+dim(ker(T))$
2. Suppose $T$ is injective, thus we get $dim(ker(T))=0$
   1. $dim(W)=dim(V)=dim(Image(T))=0$
   2. Therefore, $T$ is surjective
3. Suppose $T$ is surjective, then we get $dim(Image(T))=dim(W)$
   1. $dim(W)=dim(V)=dim(W)+dim(Ker(T))$
   2. $⟹dim(Ker(T))=0$
   3. Therefore, $T$ is injective