Linear map

Rank-nullity theorem

Let be a linear map. Then any complement of the kernel is isomorphic to the image linalg

and thus the sum of the rank and the nullity equals the dimension of ¹

In full generality, this is downstream of AC.

Proof

By ^Existence we have whence . Let . Note is monic since . Let . Since for and we have

hence so is an isomorphism It follows immediately that .

Corollaries

• It follows that an endomorphism on a finite-dimensional vector space is monic iff it is epic.

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Footnotes

1. 2008. Advanced Linear Algebra, p. 63 ↩