Intuitive explanation of why $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$ I'm having a hard time truly understanding the meaning of $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$ where $V$ is the domain of a linear transformation $T:V\to W$. I've used this equation several times in many problems, and I've gone over the proof and I believe that I fully understand it, but I don't understand the intuitive reasoning behind it. I'd appreciate an intuitive explanation of it.
Just to be clear, I do understand the equation itself, I am able to use it, and I know how to prove it; my question is what is the meaning of this equation from a linear algebra perspective.

I like to think of it as some form of conservation of dimension. If you have a linear mapping then it acts on each dimension of the domain (this is a consequence of linear mappings being completely determined by their action on any given basis of a space). 
There only two possibilities for each dimension, either it is preserved or it is compressed (i.e. taken to $\mathbf{0}$). The net dimension of the compressed portion of the domain is your nullity, i.e. the dimension of your kernel. The net dimension which is preserved is your rank, i.e. the dimension of your image space. This gives you an intuitive understanding of the rank-nullity theorem.
As a note, if you take a minute and think deeply then you'll realize this argument is essentially the same as the projections that trb456 mentioned.
