For anyone not familiar with the concept, a one-form is something that takes vectors as inputs and produces numbers as outputs. See <a href="https://en.wikipedia.org/wiki/One-form" rel="nofollow">https://en.wikipedia.org/wiki/One-form</a>
Dual spaces are an essential component of multilinear algebra, a subject that is simultaneously essential yet often ignored in undergraduate education.
Does this have anything to do with covariance and contravariance in terms on CS ? I couldn't understand from the article what those terms mean in the context of vectors spaces.<p><a href="https://en.wikipedia.org/wiki/Covariance_and_contravariance_(computer_science)" rel="nofollow">https://en.wikipedia.org/wiki/Covariance_and_contravariance_...</a>
I've got the feeling that the problems that vector spaces is trying to solve can be elegantly solved by quaternion approach [1]<p>[1]<a href="https://www.researchgate.net/publication/2130951_On_quaternionic_functional_analysis" rel="nofollow">https://www.researchgate.net/publication/2130951_On_quaterni...</a>
There is beauty in first constructing V* as the one-forms over V, which seems like a "one-way" derivation from it, and then finding through V<i></i> = V that they're actually two very-equal faces of the same coin.
The machinery described in the article is powerful and useful, but you don’t <i>need</i> 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.<p>So row vectors are the dual of column vectors. Job done!<p>Edit: I admit you need a bit more for covariance and contravariance though.