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n-dimensional vector space

Source: IMS 2014 - Day2 - Problem10

October 5, 2014

vectorgeometrygeometric transformationgroup theoryabstract algebralinear algebralinear algebra unsolved

Problem Statement

Let

V

be a

n-

dimensional vector space over a field

F

with a basis

\{e_1,e_2, \cdots ,e_n\}

.Prove that for any

m-

dimensional linear subspace

W

of

V

, the number of elements of the set

W \cap P

is less than or equal to

2^m

where

P=\{\lambda_1e_1 + \lambda_2e_2 + \cdots + \lambda_ne_n : \lambda_i=0,1\}

.

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