An endomorphism is injective iff it's surjective

up:: Linear Transformation

Given an Endomorphism , we seek to prove that it is injective if and only if it is surjective.

By the Kernel-Image Theorem, we have that

For clarity, I’ll denote the vector space of ‘s domain, and the vector space of its codomain.

Injective Surjective

Since A linear transformation is injective iff its kernel is trivial, we have that

But since , we have that , and thus is Surjective¹.

Surjective Injective

If is surjective, we have that , for which we have

for which we conclude that is Injective.

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References

• Notas sobre Álgebra Linear

Footnotes

1. Since as a subspace, if both have the same Dimension, then they must be the same space. ↩