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Linear Tranformations

  1. Jun 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Let L : M 2x2 ---> R^3 be defined by L( a b ) = (a, b+c , d)
    c d

    a. Find ker(L), is L a 1-1 function?
    b. What is the rank of L and does it map M 2x2 onto R^3

    2. Relevant equations



    3. The attempt at a solution

    a. ker(L) = (0, 0, 0),
    b. Rank= 3
    c. 4= N + 3
    N= 1 So it does map it




    Can you guys see if this is right? It is an odd number in the book
     
  2. jcsd
  3. Jun 14, 2009 #2

    Mark44

    Staff: Mentor

    This is not right, and in fact, not even close. The kernel of L consists of 2 x 2 matrices, not vectors in R3, as the vector you show is. The 2 x 2 zero matrix is included in the kernel of this problem, but there are many other matrices as well.
    What does N represent here? Convince us that you understand what you are doing. Also, unless you have a theorem to invoke, you should show that the transformation is 1-to-1 (or not) by showing that if L(M1) = L(M2), then M1 = M2.
     
  4. Jun 14, 2009 #3
    N represents the dimension of the nullspace, I am stumped
    Am I supposed to prove it my T(v+w) = T(v) +T(w) and T(cV)=cT(v)?
     
  5. Jun 14, 2009 #4
    Also the nullspace is the solution set to Ax=O, so it being 1 means there is only 1 solution set
     
  6. Jun 15, 2009 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You don't appear to know any of the definitions. The "nullspace" and "kernel" are the same thing- just different names. You say at one point that the the kernel is just {(0,0,0)} and at another that the dimension of the nullspace is 1. Those can't both be true. The "subspace" {(0,0,0)}" has dimension 0, not 1. Learn the definitions!

    In any case, since L is applied to "2 by 2 matrices" the kernel (nullspace) must be a subspace of matrices not ordered triples.

    Here, L is defined as
    [tex]L\left(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\right)= (a, b+ c, d)[/tex]

    For a matrix to be in the kernel we must have
    [tex]L\left(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\right)= (a, b+ c, d)= (0, 0, 0)[/tex]
    or a= 0, b+ c= 0, d= 0. What numbers satisfy that? What matrices are in the kernel?

    Answer that and we'll work on the rest of the problem.
     
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