This may be a stupid question, but when I was studying the rank-nullity theorem, I was wondering about the next problem:

The rank-nullity theorem states, for a linear map T: V --> W:

"dim(Im T) + dim(Ker T) = dim V"

But what happens if you, for example, have a linear map from R² to R³. Then the dimension of the image is 3 and the dimension of V is 2. But that means that the dimension of the kernel has to be -1, and that's not possible, is it?

I hope someone can help me out with this.