So I had the following question:
$T,S:V \mapsto V$ Are linear transformations
$dim(V) = n\; $, $rank(T) = n\; $, $rank(S) = k$.
Prove that $rank(T+S) \geq n-k$
Now, I know how to put many sub-conclusions to point such as:
$T$ is one to one and onto, which means $dimKer(T) = 0, \; dimIm(T) = n\; $.
Also, $dimIm(S) = k\; $, $Im(S)\subseteq \; Im(T)$
But I'm having a difficult time putting this proof together.
Also, could you please give me some tips and point of views on how to look at such questions regarding the dimension of summation \ scalar multiplication \ composition of linear transformations?