Whilst writing out revision notes, something struck me!
let be a finite dimensional real inner product space.
let be a subspace of V.
The annihilator of , denoted where is the dual space of V.
The perpendicular space of , denoted
It might already be apparent that there is some connection between these two – both involve things going to zero, whenever something is done to everything in .
To make this more explicit, first we need a theorem.
Theorem: (weaker version of) Riesz Representation Theorem
is an isomorphism of vector spaces. Restated, we can associate each linear map uniquely with a vector such that .
and hence we see this is a linear map.
thus by positive definiteness, hence is injective.
Since we have that this is an injective linear map between spaces of the same dimension, this follows by Rank Nullity. Alternatively:
take a basis
and we can check that this works.
Now to use this theorem to show the connection.
but for some .
So we have the result we wanted:
This means all of those oh-so-fun identities one derives for annihilator subspace structure can be passed over to perpendicular subspace structure!
Given the identity
we can deduce
linearity of gives us
so by injectivity of we have