dimension

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October 23rd 2010, 12:55 PM
alexandrabel90

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?
October 23rd 2010, 01:09 PM
HappyJoe

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

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

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

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

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 $f$ is implicitly assumed to be linear, so the author of the problem would probably like a linear example of a non-isomorphism.
October 23rd 2010, 01:29 PM
alexandrabel90

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?
October 24th 2010, 01:23 AM
HappyJoe

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.