 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2014/3/a/ii/solution

W \leq V^{*}

W^{\circ} = {L \in V^{**} : L (ϕ) = 0 for every ϕ \in W} .

dim W = k

, extend a basis of

W

to one of

V^{*}

. In the resulting dual basis of

V^{**}

, the last

n - k

W^{\circ}

. Hence

dim W + dim W^{\circ} = n

j : V \to V^{**}

j (x) (ϕ) = ϕ (x)

. It is a linear isomorphism in finite dimension. Its inverse image of the annihilator is

j^{- 1} (W^{\circ}) = {x \in V : ϕ (x) = 0 for every ϕ \in W} .

This identification uses evaluation only; no choice of inner product is involved.

 New to topics? Read the docs here!