Let F = {0, 1} be the field with two elements, and let · denote the standard dot product in F100, defined by (x1, ... , x100) · (yı,... , y100) = x1yı + … + x100Y100 € F. Denote by Cj, ... , CM the clubs that the town has formed under the new rules. (So that M is the number of distinct clubs.) After numbering the inhabitants from 1 to 100, associate to each C; a vector v; in F100, whose i-th entry is equal to 1 precisely if the i-th inhabitant belongs to C;. (a) Show that the vectors V1, . , UM satisfy if i + j, V¡Vj = if i = j, (b) Use the result of (a) to show that the vectors (v1, .. , UM) are linearly independent in F100. (c) Conclude that M < 100.
Let F = {0, 1} be the field with two elements, and let · denote the standard dot product in F100, defined by (x1, ... , x100) · (yı,... , y100) = x1yı + … + x100Y100 € F. Denote by Cj, ... , CM the clubs that the town has formed under the new rules. (So that M is the number of distinct clubs.) After numbering the inhabitants from 1 to 100, associate to each C; a vector v; in F100, whose i-th entry is equal to 1 precisely if the i-th inhabitant belongs to C;. (a) Show that the vectors V1, . , UM satisfy if i + j, V¡Vj = if i = j, (b) Use the result of (a) to show that the vectors (v1, .. , UM) are linearly independent in F100. (c) Conclude that M < 100.
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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Transcribed Image Text:Let F
{0, 1} be the field with two elements, and let · denote the standard dot product in F100, defined
by
(x1, ... , x100) · (yı,... , y100) = x1yı +
+ X100Y100 E F.
...
•.. 2
Denote by C1,... , Cm the clubs that the town has formed under the new rules. (So that M is the number
•.. .
of distinct clubs.) After numbering the inhabitants from 1 to 100, associate to each C; a vector v; in
F100, whose i-th entry is equal to 1 precisely if the i-th inhabitant belongs to C;.
(a) Show that the vectors V1, ..
. , UM satisfy
if i + j,
if i = j,
V¡V j
1
(b) Use the result of (a) to show that the vectors (v1, ... , UM) are linearly independent in F 100.
(c) Conclude that M < 100.
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