Results 1 to 6 of 6

Math Help - Is a Basis unique?

  1. #1
    Junior Member
    Joined
    Aug 2010
    Posts
    31

    Is a Basis unique?

    When trying to find a basis, I got (-3, -11, 1, 0 ) & (3, 1, 0, 10)

    When i saw an example using a different technique to me, (3, 1, 0, 10) & (3, 0, 1, 11) was the basis?

    Is that possible, generally and in this example? Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    May 2011
    Posts
    8
    A basis is not unique.
    As long as your basis spans the vector space, and the elements in the basis are linearly independent, then your basis will work; in general, we usually try to find the smallest or simplest basis that will work.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,397
    Thanks
    760
    Quote Originally Posted by spiral View Post
    A basis is not unique.
    As long as your basis spans the vector space, and the elements in the basis are linearly independent, then your basis will work; in general, we usually try to find the smallest or simplest basis that will work.
    all bases have the same "size" (literally if V is finite-dimensional).

    in this case, however, one of the bases must be incorrect: (3,0,1,11) is not in span{(-3,-11,1,0),(3,1,0,10)}.

    if it were, we would have (3,0,1,11) = a(-3,-11,1,0) + b(3,1,0,10), giving us:

    -3a + 3b =3
    -11a + b = 0
    a = 1
    10b = 11 --> b = 11/10. but -11 + 11/10 is not 0, so we cannot find any such a and b.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,780
    Thanks
    1522
    "Ackbeet: Deveno beat me"

    Do you have that on a rubber stamp?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2011
    Posts
    8
    -3a + 3b =3
    -11a + b = 0
    a = 1
    10b = 11 --> b = 11/10. but -11 + 11/10 is not 0, so we cannot find any such a and b.
    I think your "a" is wrong.
    I got a = 1/10, b = 11/10 which seems to work out.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,397
    Thanks
    760
    the 3rd coordinate of (3,0,1,11) is 1. the 3rd coordinate of a(-3,-11,1,0) + b(3,1,0,10) = (-3a+3b,-11a+b,a,10b) is a.

    hence, if the two are to be equal, a MUST EQUAL 1. the 4 equations

    -3a+3b = 3
    -11a+b = 0
    a=1
    10b=11

    must all have a SIMULTANEOUS solution for (3,1,0,11) to be in the span.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basis of ker L --> Basis of vector space?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: September 17th 2011, 08:57 AM
  2. Replies: 4
    Last Post: August 30th 2011, 04:48 PM
  3. Replies: 4
    Last Post: March 30th 2011, 06:40 AM
  4. unique solution
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 2nd 2011, 09:34 AM
  5. Basis and co-ordinates with respect to a basis.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 5th 2010, 07:26 AM

Search Tags


/mathhelpforum @mathhelpforum