$$\dim(U) = \dim (\ker\phi) + \dim(\text{im }\phi)$$
Hey all I'm just a little confused about the above mentioned theorem; namely, how does one measure the dimensions of a kernel? A kernel is normally defined as a set, and I don't see how I could measure the dimensions of a set.. and surely, if we were referring to a bijection in the above-mentioned linear mapping, $\phi$, then wouldn't the theorem no longer hold true at that point because the dimensions of the ker would surely be 1, as there is definitely exactly one identity-element mapped to the identitity element in the other set, right? $n = 1 + n$?
a one-to-one mapping implies a Kernel with only one element and a set with only one element has how many dimensions? it sounds like I'm comparing apples to pears..
I hope it doesn't come across as a stoopid question, thanks yo.
A kernel is not only a set, it is a subspace of $U$. Which is to say, it is itself a vector space, and as such has a notion of dimension.
It is actually a good exercise to show that this is indeed the case.