TRU Mathematics 20−1 〔1984)
ATEST OF LINBAR INDBPENDENCY OF n−VECTORS
Motosaburo MASUYAMA (Received January 13, 1984) L・t{・i}ri 一・・2・…・・ノb・…rth・g・…b・・i・・f・E…id・・・…p・c’・・ th・n a・y・t・m・f vect・r・{xi}ri−1・2・…・・ノi・linearly i・d・p・ndent・if〔・) ・4.r・・…バ・・一ヱ/・
z=ヱ [1]. . This inequality can be iiTrproved slightly as follows. With・ut 1・・s・f g・nerality w・m・y assum・th・t㌢’S・nd xt’s a「e unitvectors and
(2) ei’ =「δZl・δz2・…・δt。ノ・ where δ.、denotes Kronecker.s delta. 』 tJ ’ L・tu・assum・ ・t fi・・t th・t・・プ・are lin・arly d・p・nd・nt・nd fi・d the maxi皿um of 5. To do so , we may assume that (3) eos(ei・Xi)≧O without loss of generality. ’ ・・プssp・n・h)rP・rPlane r。・.its sub・pace・L・t the equ・ti°・°f・b・ (4) xte= O with e te=1. Obeing the origin, 1et us set〔・) …、毎 ・・d・・%鴎
and 1・t th・f・・t・f th・p・rP・ndi・ular E吉t・th・h)rPerplan・ ・ b・Gi・Th・n °°s「θs認。’1。1蕊’,㌘。論盤膓;1:°;’1:al蕊隠1’:1。。、
toE.G.. z t On the other hand, we obtain、from the equation 〔4) the relation 〔6) di=ei ’e・ which i珂plies that 56 M.MASUYAMA