Let
be a linear transformation of some linear space
to a linear space
. Let
be infinite-dimensional. Prove that at least
or
is infinite-dimensional.
Intuitively it makes sense by the rank plus nullity theorem since for a finite-dimensional linear space
we have
. Clearly, if
is infinite-dimensional then
or
must be infinite dimensional as well. Although, I can't seem to show this. Contradiction seems the best approach.
Okay, lets assume that
and
. Let
be a basis for
. Also, let
be a basis for
.
Now let
be linear independent vectors in
where
.
You can't do that: you ALREADY chose : the first k to be a basis of N(T) and the next ones such that their images under T are a basis of T(V) What you can say (and show!) is : since are lin. ind. (proof?) in V and since V is infinite dim., we can add lin. ind. elements such that...etc.
Now applying
to a linear combination of these elements we see that
since
But
such that
. From this we see that
. Which is saying a linear combination of linear independent elements is equal to a linear combination of linearly dependent elements....which is a contradiction?....I don't know what else to do from here.
Any help would be appreciated! Thank you.