A linear transformation is an isomorphism if and only if it maps basis to basis A linear transformation $\tau\in L(V,W)$ is an isomorphism if and
only if there is a basis $B$ for $V$ for which $\tau B$ is a basis for $W$. In this case, maps any basis of $V$ to a basis of $W$?
Any linear transformation of Linear Space is isomorphism, if the V, W have the same dimensionality &lt;-> transformation is not nullable &lt;-> core is rank 0 &lt;-> det L != 0 &lt;->  there exist basis of V that maps in basis of W. if its true for 1 case its true for all cases, by definition of basis.

Because of that the only real thing that defines "face" of LS is number of its dimensions. All Linear spaces of same dimension are isomorph to each other.
Any linear transformation of Linear Space is isomorphism, if the V, W have the same dimensionality &lt;-> transformation is not nullable &lt;-> core is rank 0 &lt;-> det L != 0 &lt;->  there exist basis of V that maps in basis of W. if its true for 1 case its true for all cases, by definition of basis.

Because of that the only real thing that defines "face" of LS is number of its dimensions. All Linear spaces of same dimension are isomorph to each other.