Let $V$ be a vector space over a field $F$ which is not algebraically closed and $T:V \rightarrow V$ be a linear transformation. I am just trying to understand if injectiviity can be used to study the structre of the minimal polynomial of $T$ because none of the theorems from my linear algebra textbook make this notion explicitly clear.
So suppose first that $T$ is a bijection and that $V$ has dimension $1$. Since $T$ is a bijection can we just fix a $v \in Image (T)$ so that $Tv = av$ for some $a \in F$ and therefore the minmal polynomial of T is just $x-a$. (since $V$ has dimension 1).
Question: Suppose $dim(V) \ge 2$ if $T$ is a bijection does it follow that the minmal polynomial of $T$ has a linear factor or can we say something stronger?
It is not clear to me at this point if there is a connection between injectivity of a linear operator and diagonaliziablity .