Every vector space with same dimension are isomorphic to each other

up:: Vector Space

Given vector spaces with same Dimension , we seek to prove that they are isomorphic.

Since we know that There is only one linear transformation which maps between bases of equal size, there is a Linear Transformation such that a Hamel Basis in is mapped to a basis in .

is surjective is isomorphism

Let , which can be written as the Linear Combination .

Per definition of , we have that . Since we have that , we have that . Thus, , and thus is Surjective.

Since A linear transformation between spaces of equal dimension is injective iff it’s surjective iff it’s an isomorphism, we have that is an Vector Space Isomorphism between and .

Corollaries

• Every vector space is isomorphic to the cartesian product of its field

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References

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