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Vectors in Space

Source: ILL 1970 - Problem 26.

May 24, 2011

vectorcombinatorial geometrylinear algebralinear algebra unsolved

Problem Statement

Consider a finite set of vectors in space

x=\sum_{i=1}^{n}{\lambda _i a_i}

and the set

E

of all vectors of the form

x=\sum_{i=1}^{n}{\lambda _i a_i}

, where

\lambda _i \in \mathbb{R}^{+}\cup \{0\}

. Let

F

be the set consisting of all the vectors in

E

and vectors parallel to a given plane

F

. Prove that there exists a set of vectors

y

such that

F

is the set of all vectors

y

of the form

y=\sum_{i=1}^{p}{\mu _i b_i}

, where

\mu _i \in \mathbb{R}^{+}\cup \{0\}

.

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