\[T\]
sends standard basis elements onto other standard basis element as follows:\[T(\mathbf{e}_j) = \left\{ \begin{array}{11} \mathbf{e}_{j+1} & j=0,...,n-1 \\ 0 & j=n \end{array} \right. \]
\[T(\mathbf{e}_1)=\mathbf{e}_2, \: T(\mathbf{e}_2)=\mathbf{e}_3 , \: ..., T(\mathbf{e}_{n-1})=\mathbf{e}_n , \: T(\mathbf{e}_n) =0\]
Hence
\[T^n(\mathbf{e}_1)=0\]
If
\[A\]
is the matrix representing \[T\]
then \[A^n(\mathbf{e}_1)=0\]
\[A\]
satisfies a polynomial of degree \[n\]
but cannot satisfy a polynomial of degree less than \[n\]
since \[T^k(\mathbf{e}_1)=\mathbf{e}_{k+1}\]
for \[k < n\]
.The minimum polynomial is
\[A^n\]
.