Exercise 7.A.18

Proof. We give a counterexample. Let $V={ℝ}^2$ and $T$ defined by

where $e_1,e_2$ is the standard basis of ${ℝ}^2$. Its matrix with respect to the same basis is

Taking the transpose, we see that $T^{\text{∗}}$ is defined by

Note that $\left|\right|T{e}_1\left|\right|=\left|\right|T^{\text{∗}}{e}_1\left|\right|$ and $\left|\right|T{e}_2\left|\right|=\left|\right|T^{\text{∗}}{e}_2\left|\right|$. However $M(T^{\text{∗}})M(T)\ne M(T)M(T^{\text{∗}})$, thus $T$ is not normal. □