I don't really understand this comment at the end of Boyd's Convex Optimization, Section 1.6.
In the following, $S^k$ represents the space of $k \times k$ symmetric matrices.
"We usually leave it to the reader to translate general results or statements to other vector spaces. For example, any linear function $f : R^n → R$ can be represented in the form $f(x) = c^T x$, where $c \in R^n$. The corresponding statement for the vector space $S^k$ can be found by choosing a basis and translating. This results in the statement: any linear function $f : S^k \to R$ can be represented in the form $f(X) = tr(CX)$, where $C \in S^k$."
Can anyone explain the last two sentences? Why does the $tr(CX)$ function sufficiently capture all possible linear functions on symmetric matrices?