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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
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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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.
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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