Finite dimensional vector spaces are isomorphic if and only if they have the same dimension, namely m = n.

In this problem,

Statement 1 is correct. Since the dimension m and n are not the same, L cannot be an isomorphism.

Statement 2 is false. V and W are isomorphism as long as m = n, but the linear map L may not be an isomorphism and so L(B) won’t be a basis.

Statement 3 is false, and the reason is the same as in 2.