1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear Algebra matrix linear transformation

  1. Feb 23, 2016 #1
    • Member warned about posting with no effort shown
    1. The problem statement, all variables and given/known data
    Consider the linear transformation T from
    V = P2
    W = P2
    given by
    T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2
    Let E = (e1, e2, e3) be the ordered basis in P2 given by
    e1(t) = 1, e2(t) = t, e3(t) = t^2
    Find the coordinate matrix [T]EE of T relative to the ordered basis E used in both V and W, that is, fill in the blanks below: (Any entry that is a fraction should be entered as a proper fraction, i.e. as either x/y or -x/y where x and y are positive integers with no factors in common.)
    T(e1(t)) = _e1(t) + _ e2(t) + _e3(t)
    T(e2(t)) = _e1(t) + _e2(t) + _e3(t)
    T(e3(t)) = _e1(t) + _e2(t) + _e3(t)
    and therefore :
    [T]EE = the combined matrix of the coefficients of the above three equations.

    Sorry for the poor formatting

    2. Relevant equations
    No idea

    3. The attempt at a solution
    I have no idea
    I have no idea how to do this, if someone can give me a step by step solution so that I can understand each step and process I would appreciate it greatly.
    Thanks for your help.
  2. jcsd
  3. Feb 23, 2016 #2


    User Avatar
    Science Advisor

    You mean T(a0+ a1t+ a2t^2) on the left, right?

    To find the matrix corresponding to a linear transformation in a given ordered basis, apply that linear transformation to each basis vector in turn and write the result as a linear combination of those basis vectors. The coefficients give a column of the basis. For example, to apply the linear transformation two the first basis vector, T(e1), note that e1= 1+ 0t+ 0t^2. That is, a0= 1, a1= 0, and a2= 0. Since T is defined by "T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2". T(1+ 0t+ 0t^2)= (-4(1)+ 2(0)+ 3(0))+ (2(1)+ 3(0)+ 3(0))t+ (-2(1)+ 4(0)+ 3(0))t^2= -4+ 2t- 2t^2. The first column of the matrix is [itex]\begin{pmatrix}-4 \\ 2 \\ -2\end{pmatrix}[/itex].

    Now do the same to the second and third basis vectors.
  4. Feb 23, 2016 #3
    T(f1(t)) = f1(t) + f2(t) + f3(t)
    When I put in the first one which is f1(t) = 1, so it would be 1 + 0t + 0t^2. So when I plug it in, I should get (2(1) + 0 + 0) + (2(1) + 0 + 0)t + (-2(1) + 0 + 0)t^2
    = (2,2,-2) However when I put this in it is wrong, does it have something to do with how f2(t) is now 1 + t, and f3(t) is now 1 + t + t^2?
    T(f2(t)) = f1(t) + f2(t) + f3(t)
    T(f3(t)) = f1(t) + f2(t) + f3(t)
  5. Feb 23, 2016 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    So, you have ##T(e_1) = 2 e_1 + 2 e_2 - 2 e_3##. Why do you think this is wrong?
  6. Feb 23, 2016 #5
    When I enter the numbers in it says it is wrong, except the - 2 at the end which is right. I feel like I am doing something wrong since f2(t) is now 1+t.
  7. Feb 23, 2016 #6


    Staff: Mentor

    Where did f1, f2, and f3 come from? The same basis, i.e., ##e_1, e_2,## and ##e_3## is used in both V and W. In this problem V and W are the same spaces.
    You aren't using the definition of the transformation, which I've copied below.
    ##T(a_0 + a_1t + a_2t^2) = (−4a_0 + 2a_1 + 3a_2) + (2a_0 + 3a_1 + 3a_2)t + (−2a_0 + 4a_1 + 3a_2)t^2##
    Use this definition to see what T does to ##e_1## (HallsOfIvy already did this one), ##e_2##, and ##e_3##.
  8. Feb 23, 2016 #7
    No sorry I didn't clarify. I finished the last question, this is a completely new question. I solved the previous question with hall of ivys help. However when I apply the same strategy to this question I cannot solve it. This question has f instead of e. Sorry for the confusion
  9. Feb 23, 2016 #8


    Staff: Mentor

    Then you should start a new thread, with the complete problem statement of the new problem.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook