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Range of Left Multiplication Matrix

  1. Dec 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the left multiplication L_A:R^4->R^3 corresponding the matrix A:

    1 2 -1 3 = 2
    2 4 -1 6 = 5
    0 1 0 2 = 3

    What is the rank of L_A and the Range of L_a?

    2. Relevant equations



    3. The attempt at a solution
    I have two main problems with this question. First, what is the left multiplication? I can't seem to find it anywhere and the prof said that we should know it from previous classes, but I have never heard of the left multiplication. Secondly, I understand that that the Range is supposed to be the solution space for Ax=B: is that correct?
    Thank you so much for your help in advance.
     
  2. jcsd
  3. Dec 5, 2012 #2

    Mark44

    Staff: Mentor

    Why are the = signs in there?

    This is A.
    1 2 -1 3
    2 4 -1 6
    0 1 0 2
    Edit: L_A is a transformation whose matrix representation is A. L_A takes a vector in R4 as input, and produces a vector in R3.

    The transformation L_A corresponds to A times a vector. Here A is on the left.

    No. In the equation Ax = b, the range of A is the set {b | b = Ax}

    Capital letters are usually used to represent matrices, and lower case letters are usually used to represent vectors.
     
  4. Dec 5, 2012 #3
    Oh, I see. Thank you. So for the Range, because they give you what the system of equations equals, would that be your b? or would you solve for Ax=b where b is, say, (x,y,z)?
    Thanks again.
     
  5. Dec 5, 2012 #4

    Mark44

    Staff: Mentor

    Your problem statement isn't very clear. Are they asking whether <2, 5, 3>T is in the range of L_A?

    Or are they asking what is the range of L_A?

    Your notation is confusing, because in your first equation x is a vector, while in b, x is the first component.

    To find the range, write the matrix equation Ax = b as an augmented matrix like so: [A | b], where b = <b1, b2, b3>T. Then row-reduce this augmented matrix. What you're finding is that for any vector b, is there a vector x such that Ax = b?
     
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