This question is about appropriate notation of row vectors in linear algebra.
If two vectors $\vec{a}$, $\vec{b}$ are defined and $\vec{a}$ is defined to be a row vector and $\vec{b}$ is defined to be a column vector is it then appropriate to write: $$ c = \vec{a}\vec{b} $$ for the dot product or matrix multiply of the two? While this seems possible, I don't recall ever seeing it. More broadly, what is, or is there a notation convention regarding mixing row/column vectors?
My limited experience suggests 1) all vectors are assumed to be column vectors 2) row vectors are indicated by using the transpose notation, so, the above would always be $\vec{c} = \vec{a}^T\vec{b}$ ? Thanks in advance.
As you suggest, the tacit assumption is that vectors are column vectors, with row vectors indicated via transpose. However, it's also equally okay to start with all vectors are row vectors, and work off there. Just don't mix and match.
There isn't any standard notation to indicate row vs column vectors, and thus, consistency in using all row vectors or all column vectors is quite important.
As for how you'd write the dot product of vectors as a matrix product, if they are row vectors, it would be $ab^T$, otherwise it'd be $a^Tb$. I've seen more of the former than the latter, but both are fine.