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Linear algebra-solution space

  1. Feb 2, 2013 #1
    1. The problem statement, all variables and given/known data
    a)Let M be a m*n matrix and x be a n*1 coordinate vector. How can you check whether or not x is in the solution space of M?

    [0 1 1 1 0]
    M=[1 1 0 0 1]
    [0 1 1 0 1]
    [1 0 1 0 0]

    b)To decide whether or not the following are in the solution space of M
    i) v1=[1 0 0 1 1]T ii) v2=[1 0 0 1 1]T
    *T means the transpose of the matrices

    Any help is appreciated

    2. Relevant equations


    3. The attempt at a solution

    I started by solving the homogeneous linear equation:

    [0 1 1 1 0]
    [1 1 0 0 1]*[0 1 0 1 1]T = 0
    [0 1 1 0 1]
    [1 0 1 0 0]

    [2 2 2 0]T ≠ 0

    ∴ v1 is not in the solution space of M

    Am i doing the right here??
  2. jcsd
  3. Feb 2, 2013 #2


    User Avatar
    Science Advisor

    What, exactly, was the wording of the question? To ask about the "solution space" of just a matrix, M, makes no sense. We talk about the "solution space" of an equation like "Mx= b" and the answer depends on b as well as M. The solution space of "Mx= 0", with b specfically equal to 0, is the "null space" or "kernel" of matrix M.

    Yes, to determine whether a given vector, x, is in the solution space of Mx= b, simply multiply M and the given x and see if the result is equal to b. Assuming that the problem is really asking whether the given x is in the "null space", so that b= 0, it is immediately clear that the top row times x does not give 0 and so x is not in the null space.

    But I am still concerned about the wording of the question. If it really said "solution space", it is possible that there is some non-zero "b", perhaps given in a previous part of the problem, that you missed.
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