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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?

  1. A.1
  2. B.(q^(2n) − q^n)(q^(2n) − q^(n+1))/q^n
  3. C.(q^(2n) − 1)(q^(2n) − q)⋯(q^(2n) − q^(n−1))/q^n
  4. D.(qn−1)(qn−q)/((q2−1)(q2−q))(q^n - 1)(q^n - q)/((q^{2} - 1)(q^{2} - q))✓ Correct

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, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: An injective linear operator on a vector space is surjective

More on this topic

The chapter behind this: Bases, dimension and rank — free to read

From Vector Spaces and Linear Maps › Bases, dimension, rank–nullity

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