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.

Perhaps think of it in terms of projections?  Whatever T does not project into the image must disappear; i.e. is in the kernel.  This is why it is the domain dimension that matters.  The image is an injection into the range, so it has the same dimension as the corresponding preimage in the domain.  The image of the kernel is just zero, so it is the dimension of the kernel in the domain that matters.
