1. ## dimension

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?

2. To me it's not clear exactly what you are asking.

Are you asking: If $\displaystyle f\colon V\rightarrow W$ is a linear map between vector spaces $\displaystyle V$ and $\displaystyle W$ with $\displaystyle \text{dim}(V) = \text{dim}(W)$, show that $\displaystyle f$ need not be an isomorphism?

If so, just let $\displaystyle f$ be the linear map sending all elements of $\displaystyle V$ to the zero vector of $\displaystyle W$ (which won't be an isomorphism, whenever $\displaystyle \text{dim}(V)>0$).

And why should the equations $\displaystyle \text{dim}(V) = \text{rank}(f) + \text{nullity}(f) = \text{dim}(W)$ imply that $\displaystyle \text{nullity}(f) = 0$? The map $\displaystyle f$ could easily have rank 1, and thus non-zero nullity (at least as far as $\displaystyle \text{dim}(W)>1$).

About non-linear maps not being isomorphisms of vector spaces: Well, yeah, it is very much a true statement, but I really think that the map $\displaystyle f$ is implicitly assumed to be linear, so the author of the problem would probably like a linear example of a non-isomorphism.

3. i just took it that dim W = rank W = dim V and forgot the fact that the that is only true if f is an isomorphism.

in this case, from the example that you have given,
assuming that the dimV = n and all elements in V gets sent to the zero vector in W, does it mean that nullity is n and rank is 0 thus dimW= n?

4. In the example with the map sending everything from V (with dimV = n) to the zero vector, you are right that the nullity of the map is n, and that the rank is 0. You then write "thus dimW = n", which I don't quite get. Remember that you assumed to begin with that dimV = dimW, so if dimV=n, then dimW=n is immediately true as well.