Assume that [tex] m<n [/tex] and [tex] l_1,l_2,...,l_m [/tex] are linear functionals on an n-dimensional vector space(adsbygoogle = window.adsbygoogle || []).push({});

[tex] X [/tex].

Prove there exists a nonzero vector [tex] x [/tex] [tex] \epsilon [/tex] [tex] X [/tex] such that [tex] < x,l_j >=0 [/tex] for [tex] 1 \leq j \leq m[/tex]. What does this say about the solution of systems of linear equations?

This implies

[tex] l_j(x) [/tex] [tex] \epsilon [/tex] [tex] X^\bot [/tex] for [tex] 1 \leq j \leq m [/tex] or [tex] l_j(x)=0 [/tex] for [tex] 1 \leq j \leq m [/tex]. Since it is stated in the problem that [tex] l_1,l_2,...,l_m [/tex] are linear functionals on the vector space X, [tex] l_j(x)=0 [/tex]. 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 [Broken]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Linear Functionals Inner Product

**Physics Forums | Science Articles, Homework Help, Discussion**