MathDB

26

Part of 1970 IMO Longlists

Problems(1)

Vectors in Space

Source: ILL 1970 - Problem 26.

5/24/2011
Consider a finite set of vectors in space {a1,a2,...,an}\{a_1, a_2, ... , a_n\} and the set EE of all vectors of the form x=∑i=1nλiaix=\sum_{i=1}^{n}{\lambda _i a_i}, where λi∈R+∪{0}\lambda _i \in \mathbb{R}^{+}\cup \{0\}. Let FF be the set consisting of all the vectors in EE and vectors parallel to a given plane PP. Prove that there exists a set of vectors {b1,b2,...,bp}\{b_1, b_2, ... , b_p\} such that FF is the set of all vectors yy of the form y=∑i=1pμibiy=\sum_{i=1}^{p}{\mu _i b_i}, where μi∈R+∪{0}\mu _i \in \mathbb{R}^{+}\cup \{0\}.
vectorcombinatorial geometrylinear algebralinear algebra unsolved