Could someone explain in an intuitive and if possible as simple as possible manner (ideally with at least 1 example) the relationship between covariance/contra-variance and row/col vectors ? E.g. like one can multiply e.g. a row vector by a matrix (of compatible size of course) and obtain the same result (but transposed) as multiplying the same matrix with the same vector but on the right, with that vector as a column vector this time (juste the same vector but transposed). It this also related to the contra-variance/covariance concept and if yes how? In math notation: $Ax = (x^TA^T)^T$
Also: is there a relationship with the fact that we have to multiply (a vector in standard basis) by the inverse change-of-basis-matrix to get the coordinates in the new basis, since co/contra-variance seems related to change of basis?
If have read multiple articles (wiki etc) , and understood some aspects but still not found a simple and minimal yet including the ascpects I mentioned, explanation.
Also if possible, I would like a more intuitive (in the sense 'what is the concrete purpose of?' ) covectors... I seems to be like a kind of scalar product but still I cannot understand the meaning, usage and purpose of it and all the "dual space" thing/framework.
(I am aware that there are semi-similar questions on this forum related to this subject but the answers provided to them are not satisfactory for me and I also added the relationship aspects towards inverse change-of-basis-matrix matrix.)