\textbf{Proposition 12.15 (Properties of the Symmetric Product).} \begin{enumerate}[label=(\alph*)] \item The symmetric product is symmetric and bilinear: for all symmetric tensors $\alpha$, $\beta, \gamma$ and all $a, b \in \mathbb{R}$, $$ \begin{aligned} \alpha \beta & =\beta \alpha, \\ (a \alpha+b \beta) \gamma & =a \alpha \gamma+b \beta \gamma=\gamma(a \alpha+b \beta) . \end{aligned} $$ \item If $\alpha$ and $\beta$ are covectors, then $$ \alpha \beta=\frac{1}{2}(\alpha \otimes \beta+\beta \otimes \alpha) . $$ \end{enumerate} \textbf{Exercise 12.16.} Prove Proposition 12.15.

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Exercise 12.16 (Properties of the Symmetric Product)

Proposition 12.15 (Properties of the Symmetric Product).

(a): The symmetric product is symmetric and bilinear: for all symmetric tensors $α$ , $β, γ$ and all $a, b \in ℝ$ , $\begin{aligned} αβ & = βα, \\ (aα + bβ) γ & = aαγ + bβγ = γ (aα + bβ) . \end{aligned}$
(b): If $α$ and $β$ are covectors, then $αβ = \frac{1}{2} (α \otimes β + β \otimes α) .$

Exercise 12.16. Prove Proposition 12.15.