Isomorphism

Topics: Algebra - Linear Transformation - Vector Space

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When a linear transformation $T:V\to W$ is bijective, we call it an isomorphism.

We say that $V$ and $W$ are isomorphic if there exists a $T$ between them that’s an isomorphism. We denote this by $V\cong W$.

Inverse of T §

If $T:V\to W$ is an isomorphism, then $T^{-1}:W\to V$ (the inverse) is also an isomorphism.

Dimension and Isomorphism §

Let $V,W$ be vector spaces on $\mathbb{K}$ with a finite dimension. Then, the following statements are equivalent:

1. $\dim (V)=dim(W)=n$
2. $V$ and $W$ are isomorphic

For instance, if $V$ is a vector space on $\mathbb{K}$ such that $dim(V)=n$, then $V\cong {\mathbb{K}}^n$.