Let $K$ be a symmetric matrix of dimension $n*n$. Is the following statement true:
For any vector $C$ and $D$ of dimension $n*1$, $C^T K D = D^T K C$
The above holds when I tried with simple matrices. But I do not know if it holds in general and why.
Yes, they are equal.
$$(C^T K D)^T = D^T K^T C = D^T K C$$
Since the left-hand side is a scalar (as is the right-hand side), it equals its transpose.