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.