1. The problem statement, all variables and given/known data Hi this isn't really a question but moreso 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 2. Relevant equations Here is some of the preceding material: https://imgur.com/a/acpyU 3. 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?