Spanning Sets

[jordan]

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 (v₁...v_n) be a spanning set for the vector space V and let v be any other vector in V. Show that v,v₁,...,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 v₁, v₂, v₃ and v₄ be vectors in V. Assume that {v₁, v₂, v₃} is linearly independent and that {v₁, v₂, v₃, v₄} is linearly dependent. Prove that v₄ is in Span(v₁, v₂, v₃).

 

Same as before, I can see why it's true but can't prove it. Any help?

[timo]

@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?