Difficulty understanding vector transformation law

[nabeel17]

I am having a hard time understanding vector transformations. I know that vectors must transform a certain way and that dual vectors (or covectors) transform the "opposite" way. What is strange to me is that the basis vectors transform like dual vectors and the basis dual vectors transform like vectors. So are basis vectors actually dual vectors?

 

[fresh_42]

So are basis vectors actually dual vectors?

Yes and no. In the finite dimensional case a vectorspace and its dual are isomorphic, so every vector - basis or not - can as well be seen as a dual vector. It might become clearer if you regard a vector as a column vector and its dual vector as the same written in a row.

You should have a look on this thread: https://www.physicsforums.com/threads/what-is-the-meaning-of-the-term-dual.846338/#post-5307722