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Specific Linear Map Example

  1. Oct 4, 2007 #1
    The problem statement, all variables and given/known data
    Give a specific example of an operator T on R^4 such that,

    1. dim(nullT) = dim(rangeT) and

    2. dim(the intersection of nullT and rangeT) = 1

    The attempt at a solution
    I know that dim(R^4) = dim(nullT) + dim(rangeT) = 4, so dim(nullT) = dim(rangeT) = 2.

    I also know that nullT will have 2 basis vectors and rangeT will also have 2 basis vectors (so that 1. is satisfied), and that they must have one vector in common (so that 2. is satisfied).

    I started with T(w, x, y, z) = (w, x, 0, 0), but that only does it halfway.
    I just don't know how to generate an example that satisfies both conditions.
    Any ideas?
     
  2. jcsd
  3. Oct 4, 2007 #2

    Dick

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    Science Advisor
    Homework Helper

    Try T(w,x,y,z)=(w,0,x,0). This is somewhat confusing because you usually think of null(T) and range(T) as living in different spaces. Write it in terms of a basis {e1,e2,e2,e4}. T(e1)=e1, T(e2)=0, T(e3)=e2, T(e4)=0. So null(T)=span(e2,e4). range(T)=span(e1,e2).
     
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