isomorphism

A bijective linear map; two spaces linked this way are $V \cong W$ .

A linear transformation $T:V\to W$ is an isomorphism if it is bijective (one-to-one and onto), equivalently if it has a linear inverse $T^{-1}:W\to V$ . Two vector spaces are isomorphic, written $V\cong W$ , if some isomorphism between them exists — every structural fact about $V$ (independence, spanning, dimension) translates unchanged into a true fact about $W$ . Every $n$ -dimensional real vector space is isomorphic to $\mathbb R^n$ via the coordinate map $T(\mathbf v)=[\mathbf v]_{\mathcal B}$ .