let f: V--> W. if the dim V = dim W, prove that f might not be an isomorphism.

my working:

in this case, if the dimV= rank f + nullity f= dim W,

it means that nullity f = 0.

for f to be an isomorphism, it means that it is bijective and linear.

so in this case, we need to prove that it is not linear.

let f(x) = (x)^2 + 1. then its proven.

can it be done like this?