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Linear Inependence in polynomial space

  1. Nov 25, 2008 #1
    1. The problem statement, all variables and given/known data

    Determine whether the following set is linearly independent in P4

    {1+(x2/2) , 1-(x2/2) , x+(x3/6) , x+(x3/6)}

    2. Relevant equations



    3. The attempt at a solution

    Well I know that if any of the entries in the set can be written as the linear combination of the others, then the set is NOT linearly independent. But how can I show this without just trying to put together random combinations hoping to find one that results in another entry in the set?
     
  2. jcsd
  3. Nov 25, 2008 #2
    In this situation I think it's best to see if a linear relationship has any non trivial solutions.
     
  4. Nov 25, 2008 #3

    Dick

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    You know the functions 1,x,x^2 and x^3 are linearly independent. Use that.
     
  5. Nov 25, 2008 #4
    Im not sure what youre getting at Dick. From what youve said, I can see that each entry in the set can be expressed as a linear combination of {1,x,x2,x3}, so that means they are not linearly independent? {1,x,x2,x3} is a basis for P4 i think, so is that all there is to it? Im still kind of confused about all of this.
     
  6. Nov 25, 2008 #5
    phil, in the set {1+(x2/2) , 1-(x2/2) , x+(x3/6) , x+(x3/6)}

    if you do 1*(x+(x^3/6)) - 1*(x+(x^3/6)), that equals 0. By choosing 0 for both scalars for the first two vectors, you have shown that there is a non-trivial relationship, thus the set is linearly dependent.
     
  7. Nov 25, 2008 #6

    Dick

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    To do it systematically, write down a*(1+x^2/2)+b*(1-x^2/2)+c*(x+x^3/6)+d*(x+x^3/6)=0 and see if it has a nontrivial solution. As JG89 points out that's pretty easy here, since the last two vectors are the same. a=b=0 and c=1 d=-1. If that hadn't been the case then group them by powers of x. Since 1,x,x^2 and x^3 are linearly independent, the coefficient of each must vanish. That's four equations in four unknowns.
     
  8. Nov 25, 2008 #7

    HallsofIvy

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    The set you give is trivially dependent, as JG89 said, because two of the vectors are equal. I wonder if you didn't mean x- x3/6.

    A set of vectors [itex]\left{v_1, v_2, v_3, v_4}[/itex] is independent if and only if [itex]a_1v_+ a_2v_2+ a_3v_3+ a_4v_4= 0[/itex] implies [itex]a_1= a_2= a_3= a_4= 0[/itex].
    The set [itex]{1+ x^2/2, 1- x^2/2, x+ x^3/6, x- x^3/6}[/tex] is independent if and only if [itex]a_1(1+ x^2/2)+ a_2(1- x^2/2)+ a_3(x+ x^3/6)+ a_4(x- x^3/6)= 0[/itex] for all x implies that [itex]a_1= a_2= a_3= a_4[/itex]. You can reduce that to a system of numerical equations by combining "like terms" and setting the coefficients equal to 0 or just setting x equal to 4 distinct values.
     
    Last edited: Nov 26, 2008
  9. Nov 25, 2008 #8
    Upon further inspection that last term is actually x-(x3/6) , just like you said HallsofIvy. You guys mention grouping by powers, does that mean I group the first two, which are degree 2, and the second two which are degree 3?
     
  10. Nov 25, 2008 #9

    Dick

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    There are four groups, corresponding the powers 1,x,x^2 and x^3.
     
  11. Nov 25, 2008 #10

    D H

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    Surely you mean a1=a2=a3=a4=0, not just that a1=a2=a3=a4.
     
  12. Nov 25, 2008 #11
    You may also use the equivalence between the P4 and R4. That is to say, you may construct a relation (function): l(x)=a1 + a2*x + a3*x2 + a4*x3, where x = (a1, a2, a3, a4). So in your case the set of vectors is actually equivalent to {(1,0,1/2,0), (1,0,-1/2,0), (0,1,0,1/6), (0,1,0,-1/6)}. Now it might be straightforward to determine if this group of vectors are linear independent.
     
  13. Nov 26, 2008 #12

    HallsofIvy

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    Yes, thank you for catching that. I have corrected my response.
     
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