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Change of basis help?

  1. Dec 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Problem is assuming the mapping T: P2---->P2 defined by T(a0+a1t+a2t2)=3a0+(5a0-2a1)t+(4a1+a2)t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}.
    The part that im confused on is when I go plug in the basis values T(1),T(t),and T(t^2)? I don't know how to do it?


    2. Relevant equations



    3. The attempt at a solution


    So to find T(1) its just T(1+0t+0t2)=3a0+5a0t

    To find T(t) is just T(0+a1(t)+0T2)=3(0)+(5(0)-2a1)t+(4a1+0)t^2=-2a1t+4a1t^2

    T(t^2)= T(0+0t+a2t^2)=3(0)+(5(0)-2(0))t+(4(0)+a2)t^2=a2T2

    Usually in lots of books they omit steps like these and I'm trying to figure them out. Is this a correct way?
     
  2. jcsd
  3. Dec 9, 2012 #2

    pasmith

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    Homework Helper

    This is the right idea, but to get [itex]T(1)[/itex] you take [itex]a_0 = 1[/itex], [itex]a_1 = 0[/itex], and [itex]a_2 = 0[/itex] so that [itex]T(1) = 3 + 5t[/itex]. Similarly for the other two basis vectors.
     
  4. Dec 9, 2012 #3
    Oh ok. So for T(t) just let a0=0, a1=1 and a2=0 and for T(t^2) just let a0=0,a1=0 and a2=1?

    That looks like the standard basis {e1,e2,e3}
     
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