3Duality

IB Linear Algebra

3 Duality

Duality is a principle we will find throughout mathematics. For example, in IB

Optimisation, we considered the dual problems of linear programs. Here we will

look for the dual of vector spaces. In general, we try to look at our question in a

“mirror” and hope that the mirror problem is easier to solve than the original

mirror.

At first, the definition of the dual might see a bit arbitrary and weird. We

will try to motivate it using what we will call annihilators, but they are much

more useful than just for these. Despite their usefulness, though, they can be

confusing to work with at times, since the dual space of a vector space

V

will be

constructed by considering linear maps on

V

, and when we work with maps on

dual spaces, things explode.