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 . Now applying to a linear combination of these elements we see that
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.