Linear isomorphism

Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$. A linear isomorphism from $V$ to $W$ is a bijective linear map $T \colon V \to W$. The spaces $V$ and $W$ are isomorphic, written $V \cong W$, when a linear isomorphism $V \to W$ exists.

Remarks

The inverse map $T^{-1} \colon W \to V$ of a linear isomorphism is linear. A linear map is an isomorphism if and only if it sends some basis, or each basis, of $V$ to a basis of $W$. For an $n \times n$ matrix $A$ over $F$, the map $x \mapsto Ax$ is an isomorphism of $F^n$ if and only if $A$ is invertible. Finite-dimensional spaces are isomorphic if and only if they have equal dimension.

Explanations

L1 human

A linear isomorphism is a perfect translation between two vector spaces. It matches the vectors one to one and respects addition and scaling. Isomorphic spaces have the same linear algebra, even when their vectors look different.