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Construct a matrix whose nullspace consists of all combinations [ ]

  1. Feb 11, 2013 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Construct a matrix whose nullspace consists of all combinations of (2,2,1,0) and (3,1,0,1).

    Apparently, the answer is:
    http://www.wolframalpha.com/input/?i={{1,0,-2,-3},{0,1,-2,-1}}

    2. Relevant equations
    Ax = b (where x and b are vectors and A is a matrix) (in this case, b = 0)

    3. The attempt at a solution
    As I understand it, I am supposed to find a matrix that, when multiplied with any vector in a certain vector space, I get b = 0. Assuming that I am right theoretically, I need help to continue from here in practice.

    Any help in figuring out how to do this problem would be greatly appreciated!
     
  2. jcsd
  3. Feb 11, 2013 #2

    Zondrina

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    Homework Helper

    Re: Construct a matrix whose nullspace consists of all combinations [.

    Recall that the nullspace is the span of a set of vectors (i.e those two vectors you have there ) and since your vectors are in ℝ4, you know you have at least 2 free variables right?
     
  4. Feb 11, 2013 #3

    Mark44

    Staff: Mentor

    Re: Construct a matrix whose nullspace consists of all combinations [.

    To be more specific, you want to construct a matrix that maps any linear combination of <2,2,1,0> and <3,1,0,1> to the zero vector. Presumably the matrix is 4 x 4, but that doesn't seem to be stated in your problem. These two vectors form a subspace of R4, not an entire vector space.


     
  5. Feb 11, 2013 #4

    s3a

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    Re: Construct a matrix whose nullspace consists of all combinations [.

    Yes, I do. :)

    The answer is this: http://www.wolframalpha.com/input/?i...C-2,-1}}

    What was done, though? Did they take the vectors and make them row vectors and then perform some row operations? If so, which vector got placed in which row and, what are the row operations that were performed?
     
  6. Feb 11, 2013 #5

    Mark44

    Staff: Mentor

    Re: Construct a matrix whose nullspace consists of all combinations [.

    I would create a 4 x 4 matrix with entries a through p. Use it to multiply your two given vectors to get the 4 x 2 zero matrix. Carry out the multiplication to get 8 equations in 16 unknowns.

    I haven't worked the problem, but this is how I would start it.
     
  7. Feb 11, 2013 #6

    Dick

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    Re: Construct a matrix whose nullspace consists of all combinations [.

    If a=(2,2,1,0) and b=(3,1,0,1) then every row r of your matrix has to be orthogonal to a and b. So it has to satisfy r.a=0 and r.b=0. It's in the orthogonal subspace to the subspace spanned by a and b. Get two linearly independent solutions to those equations and use them for the rows of the matrix.
     
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