Please once tell ...

this theorem is not applicable for all Linear transformation....see this example

this theorem is not applicable for all linear transformation......see an example

yes, if between two vector spaces a LT is defined and dim(V) = dim(V') and we want to check it is an isomorphism or not then in this case it is sufficient to check only one-one no ...

These are two different things . Theorem is on vector space not Linear transformation . Refer attached explanation