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 question regarding Matrices of Linear Transformations

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the matrix representations [T][itex]\alpha[/itex] and [T]β of the following linear transformation T on ℝ3 with respect to the standard basis:

    [itex]\alpha[/itex] = {e1, e2, e3}

    and β={e3, e2, e1}

    T(x,y,z)=(2x-3y+4z, 5x-y+2z, 4x+7y)

    Also, find the matrix representation of [T][itex]^{\alpha}_{\beta}[/itex]

    2. Relevant equations


    3. The attempt at a solution

    T(e1) = (2, 5, 4)

    T(e2) = (-3, -1, 7)

    T(e3) = (4, 2, 0)

    So, I got [T][itex]\alpha[/itex] = (T(e1), T(e2), T(e3))

    but for [T]β, I got [T]β=(T(e3), T(e2), T(e1))
    However, the answers in the back of the book tell me that although my order for [T]β is correct, the vectors themselves are inverted.
    ie: T(e1) = (4, 5, 2)

    Why is this? And I'm not sure how to start the second half of the question...
  2. jcsd
  3. Mar 18, 2012 #2


    User Avatar
    Science Advisor

    The columns of the matrix of a linear transformation in a given ordered basis are the coefficents of the expression of the transformation of each basis vector, in turn, written as a linear combination of that ordered basis.

    The reason I emphasised "ordered" is that in the second basis, you are given [itex]\beta[/itex] as same vectors as in [itex]\alpha[/itex], just in a different order.
    [itex]T(\beta_1)= T(e_3)= (4, 2, 0)= 0e_3+ 2e_2+ 4e_1[/itex] so the first column is
    [tex]\begin{bmatrix}0 \\ 2 \\ 4\end{bmatrix}[/tex]
  4. Mar 19, 2012 #3
    Thank you, that cleared up some of my problems!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook