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.

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