The machinery described in the article is powerful and useful, but you don’t need it to understand dual spaces. Column vectors in R^n form a vector space. Row v

The machinery described in the article is powerful and useful, but you don’t need it to understand dual spaces. Column vectors in R^n form a vector space. Row vectors map them into real numbers via standard matrix multiplication (on the left). Also vice-versa with right multiplication. So row vectors are the dual of column vectors. Job done! Edit: I admit you need a bit more for covariance and contravariance though.