Bourbaki

Exercise 9, § 9

Algebra, II

Let KK be a finite field with q elements, VV a vector space of dimension n over KK. a) Now prove that the set of sequences (x1,x2,…,xm)(x_1, x_2, \ldots, x_m) of m≤nm \leq n vectors of VV forming a free system, has cardinal equal to

(qn−1)(qn−q)⋯(qn−qm−1)(q^n - 1)(q^n - q) \cdots (q^n - q^{m-1})
(reason by induction on n). b) Deduce from a) that the cardinal of the set of linear varieties of dimension m in a projective space of dimension n over KK is equal to
(qn+1−1)(qn+1−q)⋯(qn+1−qm)(qm+1−1)(qm+1−q)⋯(qm+1−qm).\frac{(q^{n+1} - 1)(q^{n+1} - q) \cdots (q^{n+1} - q^m)}{(q^{m+1} - 1)(q^{m+1} - q) \cdots (q^{m+1} - q^m)}.

Solution

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Machine translation of the English, by gpt-5-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.