Suppose $V$ is finite-dimensional vector space and $T \in \mathcal{L}(V)$ is a linear operator. Then does there exist a vector space $V$ such that $dim(V) = dim(null) + dim(range)$ if $dim(V) = 1$, $dim(null) = 1$, and $dim(range) = 0$?
I am trying to think of a reason why this isn't possible, but not really getting anything. It seems like a simple question to answer, but I am struggling on coming up with something 🙁
Best Answer
This is the rank-nullity theorem.
By linearity of $T$, if the dimension of its range is zero, it necessary is 0, and your condition trivially satisfied.