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Linear algebra question: Orthogonal subspaces

  1. Mar 23, 2013 #1
    1. The problem statement, all variables and given/known data

    For each of the following matrices, determine a basis for each of the subspaces N(A)

    A=[3 4]
    [ 6 8]

    2. Relevant equations

    3. The attempt at a solution

    So reducing it I got [1 4/3]
    [0 0]

    I know x2 is a free variable

    I set x2 = to β

    and found my N(A)=(-4/3β,β)T

    However the book has simply (-4,3)T

    Am I incorrect?
    Last edited by a moderator: Mar 23, 2013
  2. jcsd
  3. Mar 23, 2013 #2


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    What would have happened if you had set ##x_2=3\beta##?
  4. Mar 23, 2013 #3
    We get -4B
  5. Mar 23, 2013 #4
    To find a basis for the nullspace is to find a minimal set of vector(s) that span your nullspace. If the nullspace is [itex] \binom{-4}{3} [/itex] as the book says, then any member of the nullspace can be written as a multiple of [itex] \binom{-4}{3} [/itex]. This means that the general form of a vector in N(A) is [itex] \binom{-4β}{3β} [/itex], which is equivalent to what you have. So you have written out the general form of a vector in the nullspace (assuming that your β is a free variable), whereas your book just gave the basis vector.
    Last edited: Mar 23, 2013
  6. Mar 23, 2013 #5
    Awesome thanks!
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