Linear Algebra question regarding Matrices of Linear Transformations

[psychosomatic]

Homework Statement

Find the matrix representations [T]_$\alpha$ and [T]_β of the following linear transformation T on ℝ³ with respect to the standard basis:

$\alpha$ = {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]$^{\alpha}_{\beta}$

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]_$\alpha$ = (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...

 

[HallsofIvy]

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 $\beta$ as same vectors as in $\alpha$, just in a different order.
$T(\beta_1)= T(e_3)= (4, 2, 0)= 0e_3+ 2e_2+ 4e_1$ so the first column is
$$\begin{bmatrix}0 \\ 2 \\ 4\end{bmatrix}$$

 

[psychosomatic]

Thank you, that cleared up some of my problems!