Yes. In other words: if vectors are column vectors, then covectors are row vectors, and vice versa. In this case, taking the dual is transposing your vectors,

Yes. In other words: if vectors are column vectors, then covectors are row vectors, and vice versa. In this case, taking the dual is transposing your vectors, and the transpose of a transpose gives you back the original vectors. But mathematicians try to build the most general, most abstract framework, which doesn't rely on the specifics of a concrete example (i.e. the dot product of finite-dimensional real vectors).