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Homework Help: Linear algebra - Transformation using a Matrix

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data

    let [itex]T:R^3 \rightarrow R^3[/itex]
    be a Linear map,
    and let [itex]B=\left \{ (1,1,1),(1,1,0),(1,0,1) \right \}[/itex]
    be a Basis

    and [itex](1,0,0)\in kerT[/itex]

    a & 0 & b\\
    3 & 2a & 1\\
    2c& b & a

    a. find a,b,c
    b. find a Basis for ImT

    2. Relevant equations

    3. The attempt at a solution

    as for a.

    I think that what we need to do is find a general vector (x,y,z)
    and express (1,0,0) thorugh it
    if we multiply the matrix by what we got we will have to get (0,0,0) (because (1,0,0) is in the kernel)
    but I'm not sure how to express (1,0,0).
    I think the vector (1,0,0) in the basis B is -1(1,1,1)+1(1,1,0)+1(1,0,1) and so is

    and for the matrix we have

    from the second equation:
    a= 1
    from the first
    from the third

    Am I right?
    Last edited: Jan 30, 2012
  2. jcsd
  3. Jan 30, 2012 #2


    User Avatar
    Homework Helper

    yeah look good calling the natural basis E, (1,0,0) in the B basis is as follows
    [tex](1,0,0)^T_E = (-1,1,1)^T_B [/tex]

    And as a check in E
    [tex]-1\textbf{b}_1+1\textbf{b}_2+1\textbf{b}_3 = -1(1,1,1)^T+1(1,1,0)^T+1(1,0,1)^T=(-1+1+1,-1+1+0,-1+0+1)^T =(1,0,0)^T [/tex]
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