Minimal polynomial of one value and multiplicity one implies matrix diagonalizable

Question

Let $T \in L(V,V)$ a linear operator with minimal polynomial $m_T (x) = (x- \lambda)$ where $\lambda$ is in a field $\mathbb{K}$. Show that $T$ is diagonalizable.

Any help will be appreciated. Thanks.

Answer 1

If $m_T(x) = (x - \lambda)$, then $m_T(T) = T - \lambda I = 0$. So, $T = \lambda I$. So, $T$ is a multiple of the identity operator, and is therefore diagonalizable.