# Every n Dimensional Vector Space is Isomorphic to $\mathbb{R}^n$ We wish to prove the following claim: > **Claim**: All finite dimensional vector spaces are **isomorphic** to $\mathbb{R}^n$. A full version of the proof can be found in my goodnotes linear algebra notebook. --- Date: 20220612 Links to: [Linear Algebra MOC](Linear%20Algebra%20MOC.md) [Vectors and Representation](Vectors%20and%20Representation.md) [Elements of Rn as Vectors](Elements%20of%20Rn%20as%20Vectors.md) Tags: #review References: * [Every n-Dimensional Vector Space is Isomorphic to the Vector Space R^n | Problems in Mathematics](https://yutsumura.com/every-n-dimensional-vector-space-is-isomorphic-to-the-vector-space-rn/) * [A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero | Problems in Mathematics](https://yutsumura.com/a-linear-transformation-is-injective-one-to-one-if-and-only-if-the-nullity-is-zero/) * [linear algebra - $V$ finite-dimensional vector space and isomorphic to $\mathbb{R}^n$? - Mathematics Stack Exchange](https://math.stackexchange.com/questions/2170576/v-finite-dimensional-vector-space-and-isomorphic-to-mathbbrn) * Good notes derivation (Linear Algebra notebook) * [linear algebra - Vector spaces of the same finite dimension are isomorphic - Mathematics Stack Exchange](https://math.stackexchange.com/questions/1025926/vector-spaces-of-the-same-finite-dimension-are-isomorphic) *