Linear Algebra question regarding Matrices of Linear Transformations

  • #1

Homework Statement



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]

Homework Equations



None

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...
 

Answers and Replies

  • #2
HallsofIvy
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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]
 
  • #3
Thank you, that cleared up some of my problems!
 

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