[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.4

If $\dim \scrH=3$, then $\dim \otimes^3\scrH =27$, $\dim \wedge^3\scrH =1$ and $\dim \vee^3\scrH =10$. In terms of an orthonormal basis of $\scrH$, write an element of $(\wedge^3\scrH )\oplus \vee^3\scrH)^\perp$.

Solution. Let $e_1,e_2,e_3$ be an orthonormal basis of $\scrH$, then $$\bex e_1\otimes e_1\otimes e_1-e_1\otimes e_1\otimes e_2\in (\wedge^3\scrH )\oplus \vee^3\scrH)^\perp. \eex$$