• 検索結果がありません。

A TEST OF LINEAR INDEPENDENCY OF n-VECTORS

N/A
N/A
Protected

Academic year: 2021

シェア "A TEST OF LINEAR INDEPENDENCY OF n-VECTORS"

Copied!
2
0
0

読み込み中.... (全文を見る)

全文

(1)

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 unit

vectors 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「θ

s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 5

(2)

6        M.MASUYAMA

〔7). ・’=「d、・d2・・…d。ノ・

     Then we have

(・〕 ・一・a−d、2/1/2.

which is maxi皿al under the condition

(・). e’・ 一 d、2品22・…・d。2 一・,

iff

〔・・) d、−d、一…−dn一ガヱ/2.

     Therefore we obtain the inequality (・・) ・m、、C・n(・一・/・)ヱ/2, P・・vid・d th・t・・蠕・a・・1i・・a・1y d・p・nd・nt・1・・ther w・rd・, if

〔12〕 S・nrヱ.W2/2,

then xi’s are linearly independent.      N°te that(i)we have (12〕 nr・.・/。」ヱ/2・。.ヱ/2、 and(ii〕σis the t「ace°f the mat「i・「・・ヱ…2・…・x。ノ・wh・⇒.・are mi・ vectors. REFERENCE . [1]  Voyevodin, V.V.〔1983〕:Lineav、4 Zgeb?a, Mir. p.108.       DEPARTト正iNT OF APPLIED MATHEMATICS        SCIENCE UNIVERSITY OF TOKYO

参照

関連したドキュメント

3.1, together with the result in (Barber and Plotkin 1997) (completeness via the term model construction), is that the term model of DCLL forms a model of DILL, i.e., a

Theorem 1.3 (Theorem 12.2).. Con- sequently the operator is normally solvable by virtue of Theorem 1.5 and dimker = n. From the equality = I , by virtue of Theorem 1.7 it

Keywords and Phrases: moduli of vector bundles on curves, modular compactification, general linear

Along with the ellipticity condition, proper ellipticity and Lopatinsky condition that determine normal solvability of elliptic problems in bounded domains, one more

Now it makes sense to ask if the curve x(s) has a tangent at the limit point x 0 ; this is exactly the formulation of the gradient conjecture in the Riemannian case.. By the

This paper is a sequel to [1] where the existence of homoclinic solutions was proved for a family of singular Hamiltonian systems which were subjected to almost periodic forcing...

This set will be important for the computation of an explicit estimate of the infinitesimal Kazhdan constant of Sp (2, R) in Section 3 and for the determination of an

Note that various authors use variants of Batanin’s definition in which a choice of n-globular operad is not specified, and instead a weak n-category is defined either to be an