Is what's a vector and covector subjective?

Question

From what I understand, if you have $v \cdot w$ (or in general $v\otimes w=k$ where k is an element of a field), then either v dot or dot w can be considered as a linear functional and therefore a covector, and the other a vector. I've heard of a vector itself being called a convector, as if it is because of its contstruction or something.

Answer 1

"Subjective" is not quite the right word, but it is relative.

That is, it makes only makes sense to talk about whether something "is" a vector, or covector, or more general tensor, with respect to a base vector space $V$ which (by definition) we take to be the "vectors." Then covectors, or more precisely covectors with respect to $V$, are elements of the dual vector space $V^{\ast}$ and so forth. But it is equally true to say that the elements of $V$ are covectors with respect to $V^{\ast}$.