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Homework Help: Linear map w/ matrix

  1. Feb 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider the map L from the space of 2x2-matrices to R given by:

    L([a b]) = a+ d
    ([c d])

    For clarity, thats L(2x2 matrix) = a + d

    3. The attempt at a solution

    Im confused how any function of a matrix could possibly equal addition of two scalars, and thus have no idea where to begin.
     
  2. jcsd
  3. Feb 3, 2010 #2

    Mark44

    Staff: Mentor

    What's the entire problem? Your problem statement is incomplete.
     
  4. Feb 3, 2010 #3

    Mark44

    Staff: Mentor

    To answer your question, this transformation takes a 2x2 matrix as input, and produces a single number as output. It takes the entries in the upper left and lower right corner and produces their sum as its output.
     
  5. Feb 3, 2010 #4
    Sorry not sure how I missed that. The question is "Is L a linear map?"

    I think I MAY have it now. I need to show linearity by proving L(a+b+c+d)=L(a) + L(b) + L(c) + L(d)

    So, L(a+b+c+d) = a 2x2 matrix where each corner is (a + b + c + d)

    L(a) is a 2x2 matrix w/ each corner containing a, L(b) is a 2x2 matrix w/ each corner containing b, and same thing for L(c) and L(d). These two things do in fact equal eachother, so its a linear map.
     
  6. Feb 3, 2010 #5

    Mark44

    Staff: Mentor

    I agree that L is linear, but not the way you did it.

    What you wrote "So, L(a+b+c+d) = a 2x2 matrix where each corner is (a + b + c + d)
    " doesn't make any sense. a, b, c, and d are real numbers, the entries in a 2x2 matrix. a + b + c + d is a single real number. The domain of L is not real numbers.
    So L(a + b + c + d) doesn't make any sense, nor is L(whatever) = a 2x2 matrix. L(whatever) is a number.

    Also, how can each corner be (a + b + c + d)?

    What you want to do is something like this:
    Let A and B be 2x2 matrices.
    Now show that L(A + B) = L(A) + L(B).
    The use of cap letters for matrices prevents confusion with a and b that represent entries in a matrix.
     
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