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Vector space problem

  1. Mar 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Let V be a vector space over a field F and let x1,.....xn [tex]\in[/tex]V.Suppose that x1,.....xn form a maximal linearly independent subset of V. Show that x1,.......xn form a minimal spanning set of V.

    2. Relevant equations



    3. The attempt at a solution
    I knew that x1,....xn are linear independent and for every x[tex]\in[/tex]V the n+1 vectors x1,....xn , x are linear dependent
    then x span x1,.....xn
    i dont how to continue
    any help ?
     
  2. jcsd
  3. Mar 14, 2009 #2

    lanedance

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    Homework Helper

    hey ak123456

    first you have only n vectors

    what are your defintions for maximally lineraly independent and minimal span? always a good place to start

    to get you started though, the my reasoning would be as follows:
    as {x1,...,xn} is maximal lineraly independent show any vector in V can be written as combination of xn's, so {x1,...,xn} spans V.
    then try and show if you remove an xi, the remaining vectors no longer span V...
     
  4. Mar 15, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If {x1, ..., xn} did not span the set, there must exist some x which cannot be written as a combination of the {x1, ..., xn}. What does that tell you about {x1, x2, ..., xn, x}?

    If it were not a minimal spanning set, then there must exist a smaller set, {y1, y2,... yn-1} which did span the set. That would mean you could write each of x1, x2, ... xn in terms of those y's. What does that tell you about the independence of x1, x2, ..., xn?
     
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