 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/b/b2/e/solution

There are

p^{n}

choices for the translation. Choose the columns of an invertible matrix successively: the first is any nonzero column, the second avoids its span, and at stage

j + 1

one avoids a

j

-dimensional span of size

p^{j}

. Consequently

∣ G ∣ = p^{n} j = 0 \prod n - 1 (p^{n} - p^{j}) .

The successive linear independence conditions are exactly the criterion that the columns form a basis.

 New to topics? Read the docs here!