jordan Posted October 26, 2006 Share Posted October 26, 2006 I'm having a little trouble with some of my homework here and need a little guidence. The problems are both proofs and I have a lot of trouble deciding what a conclusive proof is. I'm looking for a little hint on what might be a good first step or two in showing the following two things: 1) Let (v_{1}...v_{n}) be a spanning set for the vector space V and let v be any other vector in V. Show that v,v_{1},...,v_{n} are linearly dependent. I'm just having trouble conclusively showing in this one that the coefficient on v must be some combination of the others. I know it's true through common sense but I can't prove it. 2) Let V be a vector space. Let v_{1}, v_{2}, v_{3} and v_{4} be vectors in V. Assume that {v_{1}, v_{2}, v_{3}} is linearly independent and that {v_{1}, v_{2}, v_{3}, v_{4}} is linearly dependent. Prove that v_{4} is in Span(v_{1}, v_{2}, v_{3}). Same as before, I can see why it's true but can't prove it. Any help? Link to comment Share on other sites More sharing options...

timo Posted November 1, 2006 Share Posted November 1, 2006 @1) There is a set [math] \{ a_i \}: v = \sum_i a_i v_i [/math] by the definition of a spanning set => [math] 0= 1*v - \sum_i a_i v_i [/math] which is linear dependence (factor 1 explicitely written for clarification). @2) there is an a>0 so that 0 = a*v4 + x*v1+y*v2+z*v3 => v4 = (x*v1+y*v2+z*v3)/a. Left for you: Why is there such an a? what if v4=0? Link to comment Share on other sites More sharing options...

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