2.3 (Adapted from [38].) We know that if

k

, then the vectors

a_(1)

,

a_(2),dots,a_(k)inR^(n)

are linearly dependent; that is, there exist scalars

\\\\alpha _(1),dots

,

\\\\alpha _(k)

such that at least one

\\\\alpha _(i)!=0

and

\\\\Sigma _(i)=1\\\\alpha _(i)a_(i)=0

. Show that if

k>=n+

2 , then there exist scalars

\\\\alpha _(1),dots,\\\\alpha _(k)

such that at least one

\\\\alpha _(i)=0,\\\\Sigma _(i)=1

\\\\alpha _(i)a_(i)=0

, and

\\\\Sigma _(i)=1^(k)\\\\alpha _(i)=0

.\ Hint: Introduce the vectors

\\\\bar (a) _(i)=[1,a_(i)^(TT)]^(TT)inR^(n+1),i=1,dots,k

, and use the fact that any

n+2

vectors in

R^(n+1)

are linearly dependent.

2.3 (Adapted from [38].) We know that if kn+1, then the vectors a1, a2,,akRn are linearly dependent; that is, there exist scalars 1,, k such that at least one i=0 and i=1iai=0. Show that if kn+ 2 , then there exist scalars 1,,k such that at least one i=0,ii=1 iai=0, and ki=1i=0. Hint: Introduce the vectors ai=[1,a]Rn+1,i=1,,k, and use the fact that any n+2 vectors in Rn+1 are linearly dependent