>>11914430Basically linear algebra concepts should come in pairs: a computational/matrix form, and a coordinate free/vector space interpretation.
Duality gives you an abstract interpretation for the operation of transposition and for row vectors that is compatible with the usual interpretation of matrices and column vectors.
Another point is that it lets you look at vector spaces «from the outside». Typically, you look at vector spaces as spanned by bases. Dually, you can look at them as being cut out by hyperplans. Duality for finite-dimensional spaces tells you that all of this is nice and well-behaved: p linearly independent equations define a codimension p
subspace.
This result is pretty much all there is to finite-dimensional duality, but duality can be useful conceptually nonetheless.
Basically it is the appropriate language when you think of linear algebra in terms of hyperplanes, which can be useful eg. when studying convex geometry.
It is also very useful to have in mind when studying bilinear algebra (a non-degenerate bilinear form being nothing else than an isomorphism between a space and its dual).
Finally, it also prepares you for the study of duality in infinite-dimensional spaces, which is no trivial matter
