Exercise 5.4.32

As Hint, we induct on the dimension $n$ of the space. For $n=1$, every matrix representation is an upper triangular matrix. Suppose the hypothesis is true for $n=k-1$. Now we consider the case $n=k$. Since the characteristic polynomial splits, $T$ must has an eigenvector $v$ corresponding to eigenvalue $\lambda$. Let $W=\mathrm{span}(\{v\})$. By Exercise 1.6.35 we know that the dimension of $V∕W$ is equal to $k-1$. So we apply the induction hypothesis to $\overline{T}$ defined in Exercise 5.4.27. This means we may find a basis

for $V∕W$ such that ${[\overline{T}]}_{\alpha }$ is an upper triangular matrix. Following the argument in Exercise 5.4.30, we know that

is a basis for $V$. And we also know the matrix representation is

which is an upper triangular matrix since ${[\overline{T}]}_{\alpha }$ is upper triangular.