Let Fq be a finite field with q elements. For nge2, let A be a 2ntimes2n matrix with entries in Fq such that rank(A)=n. Let W=vinFq^2nmid Av=0. Which of the following is necessarily the number of (n+2)-dimensional subspaces of Fq^2n that contain W?

(D) dfrac(q^n-1)(q^n-q)(q^2-1)(q^2-q)

Q27Section B (Part B)27 of 120

Linear algebra — subspaces containing a fixed null space

Let $\mathbb{F}_q$ be a finite field with $q$ elements. For $n\ge2$ , let $A$ be a $2n\times2n$ matrix with entries in $\mathbb{F}_q$ such that $\operatorname{rank}(A)=n$ . Let $W=\{v\in\mathbb{F}_q^{2n}\mid Av=0\}$ . Which of the following is necessarily the number of $(n+2)$ -dimensional subspaces of $\mathbb{F}_q^{2n}$ that contain $W$ ?

Pick an option to check your answer.

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