Nicholas Funari Voltani

A linear transformation between spaces of equal dimension is injective iff it's surjective iff it's an isomorphism

1 min read

up:: Linear Transformation

Let a linear transformation between Vector Space with equal Dimension .

We seek to prove that

  • A) Bijectiveness
  • B) Injectiveness
  • C) Surjectiveness

It is trivial that and that .

Injective Bijective

By the Kernel-Image Theorem, we have that

Thus, the dimension of the codomain is equal to the dimension of ‘s image ─ which implies that is Surjective, and thus Bijective.

Surjective Bijective

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

for which is Injective, since A linear transformation is injective iff its kernel is trivial.

Corollaries

References

  • Um Curso de Álgebra Linear, Flávio Ulhoa Coelho & Mary Lilian Lourenço. Editora EDUSP.

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