Matrix Image and Kernel

  1. 1. The problem statement, all variables and given/known data
    i) Find the Image and Kernel of A = (2,1)(-4,-2) (where each bracket is a row).
    ii) Calculate A2 and use i) to explain your result.


    2. Relevant equations
    None

    3. The attempt at a solution
    So I can do everything up to the very last bit (i think anyway).
    i) The Kernel = (1,-2) = Image.
    ii) A2 = 0 but this is where I don't know what to say. How do I use part i) to explain the 0 matrix found?
     
  2. jcsd
  3. Mark44

    Staff: Mentor

    For i, you have the basic idea, but it needs some refinement. The kernel is not just one vector; there are an infinite number of vectors in the kernel of A. Likewise, the range is not just one vector either.

    Geometrically, A maps any vector along the line 2x + y = 0 to the zero vector. A maps any vector x not along the the line 2x + y = 0 to a vector along this line. IOW, if x is not in the kernel of A, A projects it onto this line.

    For ii, since A2x = A(Ax)think about what A does to a vector x, and then think about what A does to a vector Ax.
     
  4. OK i think i understand what you mean about the kernel. so applying to to another question, if I have the matrix:

    A=[{1,0,-2},{2,2,0},{0,3,6}] and I wanted to find the kernel, I'd reduce it down to:

    A=[{1,0,-2},{0,1,2},{0,0,0}] and thus the kernel is:

    Ker[A] = f[1,-1,2] where f is any number.

    Is that about right?
     
  5. vela

    vela 12,550
    Staff Emeritus
    Science Advisor
    Homework Helper
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    I think you got the 1's and 2's switched, i.e.

    [tex]\textrm{Ker}[A] = \{\vec{x}\in \Re^3\,|\, \vec{x} = f(2,-2,1), f \in \Re\}[/tex]
     
  6. HallsofIvy

    HallsofIvy 40,367
    Staff Emeritus
    Science Advisor

    The point here is that the image is the kernel! For any vector v, Av is in the kernel of A so A(Av)= 0.
     
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