Matrix of a Linear Transformation Example

[~Sam~]

Homework Statement

Hi this isn't really a question but more so understanding an example that was given to me that I not know how it came to it's conclusion. This is a question pertaining linear transformation for coordinate isomorphism between basis.
https://imgur.com/a/UwuAC

Homework Equations

Here is some of the preceding material:
https://imgur.com/a/acpyU

The Attempt at a Solution

I'm not sure how it came to it's conclusion for the basis G, for example the vector [1,0,0] doesn't yield the same result as the standard basis, but gets [1,-2,2] instead even though it is a standard basis e1. I thought for [1,1,0] in G the result would be [1,1,2] but instead it's [-1,1,0]. I've tried working backwards from the solution but that hasn't helped. Can anyone help me understand this?[/B]

 

[Samy_A]

First thing, do you know how to compute the matrix representing $C_G$?

Once you have the matrix for $C_G$, you can compute (for example) $C_G(T(1))=C_G\begin{pmatrix}1\\0\\2 \end{pmatrix}$ and see why it "doesn't yield the same result as the standard basis".