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Kernel, Image

  1. Dec 2, 2014 #1
    1. The problem statement, all variables and given/known data
    Find Kernel, Image, Rank and Nullity of the matrix
    1 −1 1 1 
    | 1 2 −1 1 |
    0 3 -2 0 

    2. Relevant equations

    3. The attempt at a solution
    I have reduced the matrix into rref of
    3 0 1 3
    0 3-2 0
    0 0 0 0
    But am struggling to find the column vectors which are the solutions of that, which I believe is the kernel
    And then I know that the rank and nullity are the dimensions of the image and kernel respectively
  2. jcsd
  3. Dec 2, 2014 #2


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    Science Advisor

    The "kernel" of matrix A is the set of all vectors v such that Av= 0. If you write vector v as (a, b, c, d) then any vector in the kernel must satisfy a- b+ c+ d= 0, a+ 2b- c+ d= 0, and 3b- 2c= 0. The "image" is the set of all vectors, v, such that Ax= v for some vector x. If we write vector x as (a, b, c, d) then v must be (x, y, z) such that a- b+ c+ d= x, a+ 2b- c+ d= y, and 3b- 2c= z for some numbers a, b ,c, and d.
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