Part BCSIR NET December 2025count-subspaces-of-the-quotient-by-w-not-of-the-whole-space

Count subspaces of the quotient by w not of the whole space

Let F_q be a finite field with q elements. For n ≥ 2, let A be a 2n × 2n matrix with entries in F_q such that rank(A) = n. Let W = {v ∈ F_q^(2n) ∣ Av = 0}. Which of the following is necessarily the number of (n+2)-dimensional subspaces of F_q^(2n) that contain W?

A.1
B.(q^(2n) − q^n)(q^(2n) − q^(n+1))/q^n
C.(q^(2n) − 1)(q^(2n) − q)⋯(q^(2n) − q^(n−1))/q^n
D. $(q^n - 1)(q^n - q)/((q^{2} - 1)(q^{2} - q))$ ✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Count subspaces of the quotient by w not of the whole space

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