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Proving a basis

  1. Sep 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Let {e1,e2,e3} be a basis for the vgector space V over the field F.
    Put f1 = -e1, f2 = e1+e2 and f3 =e1 + e3

    Prove that {f1,f2,f3} is also a basis for V


    2. Relevant equations



    3. The attempt at a solution


    I made e1,e2,e3 be the unit bases.

    1
    0
    e1= 0


    0
    0
    e2= 1



    0
    0
    e3= 1


    which makes {f1,f2,f3}


    -1
    0
    f1= 0


    1
    1
    f2= 0


    1
    0
    f3= 1



    so {f1,f2,f3} =

    -1 1 1 0
    a1 * 0 + a2 * 1 + a3 * 0 = 0
    0 0 1 0

    What i did was set this matrix equal to zero and solve.

    The solution was only the trivial solution.
    As this proves that {f1,f2,f3} is linear independent is that enough to show that it also is a basis ?

    regards
     
  2. jcsd
  3. Sep 11, 2009 #2

    Mark44

    Staff: Mentor

    You're given an arbitrary vector space V over some field K, so you can't assume that e1, e2, and e3 are the standard basis vectors for R3. Your set {f1, f2, f3} is a basis for V iff this set is linearly independent and spans V.

    For the first part, show that the equation c1f1 + c2f2 + c3f3 = 0 has exactly one solution. For the second part, show that any vector v in V can be written as a linear combination of the vectors f1, f2, and f3.
     
  4. Sep 11, 2009 #3

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    What is your definition for basis? And why are you assuming that [itex]e_i[/itex] are unit vectors?...They can have any length and still form a basis.
     
  5. Sep 12, 2009 #4
    For the first part show that c1f1 + c2f2 + c3f3 = 0

    is the equation to prove

    c1(-e1) + c2(e1+e2) + c3(e1 + e3) = 0

    regards
     
  6. Sep 12, 2009 #5

    Mark44

    Staff: Mentor

    No. You need to prove that c1(-e1) + c2(e1 + e2) + c3(e1 + e3) = 0 has exactly one solution for the constants c1, c2, and c3.

    Rewrite this equation so that it is ___e1 + ___e2 + ___e3 = 0 (you need to fill in the blanks). Use what you know about e1, e2, and e3 being a basis for V.
     
  7. Sep 12, 2009 #6
    So I have:

    f1 = -e1
    f2 = e1+e2
    f3 = e1+e3

    I need to try and eliminate e2 and e3

    so


    f1 = -e1
    f2+f3 = 2e1+e2+e3
    f3+f2 = 2e1+e3+e2


    Then

    f1 = -e1
    f2+f3 = 2e1+e2+e3
    -1*(f3+f2) = -2e1 -e3 -e2



    f1 = -e1
    (f2+f3)+(-1*f3+f2) = 0

    f1 = -e1
    (f2+f3)+(-1*f3+f2)+f1= -e1

    (f2+f3)+(-1*f3+f2)+f1= f1


    (f2+f3)+(-1*f3+f2)+ (f1-f1)= 0

    showing that f1,f2,f3 are independant
     
  8. Sep 12, 2009 #7

    Mark44

    Staff: Mentor

    And how exactly does your work show this? Look at what I outlined for you in post 5.
     
  9. Sep 12, 2009 #8
    HI Guys
    In my lecture notes it says.

    a set S = {U1,U2...Un} of vectors is a basis of V if every v an element of V can be written uniquely as a linear combination of the basis vectors.

    I am told that:

    Let {e1,e2,e3} be a basis for the vector space V over the field F.

    Can I assume that they are lineary independent and therefore only equal to zero if all coefficients are zero?


    so using "a set S = {U1,U2...Un} of vectors is a basis of V if every v an element of V can be written uniiquely as a linear combination of the basis vectors."


    don't I just have to show that each vectore {f1,f2,f3} can be written as an linear combination of the basis vectors.

    eg. f2 = e1+e3 = 1*e1 + 0*e2 + 1*e3


    You said:
    Rewrite this equation so that it is ___e1 + ___e2 + ___e3 = 0 (you need to fill in the blanks). Use what you know about e1, e2, and e3 being a basis for V.


    As e1, e2, and e3 is a basis they are Linearly independant and span the the vector Space. So can only equal zero if all coefficients are zero
     
  10. Sep 12, 2009 #9
    The vectors that form a basis are linearly independent.


    You need to show that any vector v in V can be written as a linear combination of {f1, f2, f3} - this shows that the set generates V
     
  11. Sep 13, 2009 #10
    So say you got the vector e1

    It can be generated by

    e1 = -1*f1 + 0*f2 + 0*f3

    And as The vectors e1,e2,e3 are independent so are f1 , f2 , f3

    regards
     
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