Every vector space with same dimension are isomorphic to each other

up:: Vector Space

Given $U,V$ vector spaces with same Dimension $\dim U=\dim V=n$, 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 $T:U\to V$ such that a Hamel Basis in $U$ is mapped to a basis in $V$.

$T$ is surjective $⟹$ $T$ is isomorphism

Let $v\in V$, which can be written as the Linear Combination $v={\sum }_i{\alpha }_i{v}_i$.

Per definition of $T$, we have that $v_i=T(u_i)$. Since we have that ${\sum }_i{\alpha }_i{u}_i\equiv u\in U$, we have that $T(u)=v$. Thus, $\forall v\in V:v\in ImT$, and thus $T$ 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 $T$ is an Vector Space Isomorphism between $U$ and $V$.

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.