Linear Functionals Inner Product

[wurth_skidder_23]

Assume that m<n and l_1,l_2,...,l_m are linear functionals on an n-dimensional vector space
X.

Prove there exists a nonzero vector x \epsilon X such that < x,l_j >=0 for 1 \leq j \leq m. What does this say about the solution of systems of linear equations?This implies
l_j(x) \epsilon X^\bot for 1 \leq j \leq m or l_j(x)=0 for 1 \leq j \leq m. Since it is stated in the problem that l_1,l_2,...,l_m are linear functionals on the vector space X, l_j(x)=0. Does this reasoning even help me find the proof? I am stuck.

If you have trouble reading this, it is also at http://nirvana.informatik.uni-halle.de/~thuering/php/latex-online/olatex_33882.pdf

 

[StatusX]

I don't know what X^\perp means, nor why you are taking the inner product rather than just applying the functional to the vector. You'll obviously need to involve n and use n>m, and the easiest way to use this is appeal to the existence of a basis with n elements.

 

[wurth_skidder_23]

so basically I'm trying to prove that for some nonzero x, l_j(x)=0?

 

[StatusX]

Yea, for some x, for all j.