1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Splitting a vector into a rowspace component and a nullspace component

  1. Nov 1, 2011 #1
    1. The problem statement, all variables and given/known data
    Find a basis for the orthogonal complement of the row space of A:
    A =
    [1 0 2
    1 1 4]

    Split x = (3,3,3) into a row space component xr and a nullspace component xn.


    3. The attempt at a solution
    For the first part of the problem I took A to RREF
    R =
    [1 0 2
    0 1 2]
    and then solved to find a basis for the nullspace, x * (-2,-2,1), which should be the basis for the orthogonal complement of the row space.

    I THINK that's right but maybe it's not because I'm stuck on the second part, splitting x = (3,3,3). Do I try to find a vector in Row(A) and a vector in Nul(A) that, when added, produce the vector (3,3,3)? How do I solve for that?

    Help is very much appreciated!

    Also, what's the best way to represent a matrix on these message boards?
     
  2. jcsd
  3. Nov 1, 2011 #2
    nevermind, I figured it out
     
  4. Nov 1, 2011 #3

    Deveno

    User Avatar
    Science Advisor

    an observation:

    x(-2,-2,1) isn't the basis for Null(A), it's the entire space. "a" basis, is any particular vector, using a non-zero value for x. x = 1 works well, so one basis is:

    {(-2,-2,1)}.

    to find xn and xr, we're looking for a,b, and c with:

    (3,3,3) = a(-2,-2,1) + (b(1,0,2) + c(1,1,4))

    the first term is in Null(A) and the second sum of two terms in in Row(A).

    this isn't that hard, we have:

    (3,3,3) = (b-2a+c,c-2a,a+2b+4c), or, if you prefer:

    [tex]\begin{bmatrix}-2&1&1\\-2&0&1\\1&2&4\end{bmatrix} \begin{bmatrix}a\\b\\c\end{bmatrix} = \begin{bmatrix}3\\3\\3\end{bmatrix}[/tex]

    solve this for a,b and c.

    ****

    the string i entered to input the matrix equation was:

    [tex]\text{[tex]\begin{bmatrix}-2&1&1\\-2&0&1\\1&2&4\end{bmatrix} \begin{bmatrix}a\\b\\c\end{bmatrix} = \begin{bmatrix}3\\3\\3\end{bmatrix}\[/tex]}[/tex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Splitting a vector into a rowspace component and a nullspace component
  1. Component of a vector (Replies: 1)

Loading...