Why does dim Range T* = dim Range T?

Question

Suppose T $\in$ $\mathrm L$(V,W)
So far I have, dim W = dim range T* + dim null T* and range T*= (null T)$^\bot$
So taking the orthogonal complement of dim W, then you have dim V = dim null T + dim range T. This is where I got stuck.

Answer 1

You'll also need the fact that $$\operatorname{dim} \operatorname{null} T + \operatorname{dim} (\operatorname{null} T)^\perp = \operatorname{dim} V.$$ Putting it all together yields the result.