Problem 4.66. Compute the number of blocks containing each individual point in a finite projec- tive plane build over Fg.
Problem 4.66. Compute the number of blocks containing each individual point in a finite projec- tive plane build over Fg.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve 4.66. The definition and collary is to help answer problem. Thank you!
Transcribed Image Text:Problem 4.66. Compute the number of blocks containing each individual point in a finite projec-
tive plane build over Fq.
Transcribed Image Text:Definition 4.15. A basis is a spanning set of minimum size in a vector space. Normally, bases
are not ordered. An ordered basis for a vector space is a basis with the members of the basis
presented in a particular order.
Lemma 4.15. Let F, be a finite field with q elements. Then the vector space F" has
n-1
II" – g*)
k=0
ordered bases.
Proof:
We count the ordered bases by selected the members of the basis in order. There are q" vectors in
vector space, one of which is the zero vector. The first member of the ordered basis is one of the
qn – 1 non-zero vectors. This vector generates a one-dimensional subspace containing q vectors.
The second element is chosen from outside of this subspace, giving us q" – q choices. In general
the first k vectors chosen span a subspace of size qk leaving q" – qk choices for the next vector in
the basis. The basis must have n elements and the choices made are, other than the constraint
of being outside of the subspace already covered, independent, and so they multiply. The formula
given in the lemma follows. O
Corollary 12. Adopt the situation in Lemma 4.15. There are
т-1
II (7" – *)
k=0
ordered bases for m-dimensional subspaces of F".
Proof:
The corollary follows because it simply stops the basis selection process early. O
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