> About dual vectors; I haven't read about dual vectors as of yet.

If you wish you can think of the dual of a vector (a one-form) as a
one row matrix (and the vector as a one column matrix).

If you have a scalar product operation defined there is a canonical
way to get a dual of a vector.

For the vector v and scalar product <,>, the oneform for the vector is
v' such that v'(v) = <v, v>

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