Possible Error in Chapter 2 [base change,2.102]

[phle86]

Describe the mistake
I think there is a mistake regarding the formula/example 2.102 --> Example 2.23

Location
Please provide the

1. version (bottom of page)
2. Chapter 2
3. page 42
4. 2.102

Proposed solution
The transformation matrix A [(2,1),(1,2)] regarding the canonical basis [(1,0),(0,1)] should be changed in respect to the new basis [(1,1),(1,-1)] --> The listed transformation matrix A[(3,0),(0,1)] is in my opinion wrong. Should it not be [(3/2,1/2),(3/2,-1/2)] ?

Additional context
Maybe i dont understand something? Could you please provide a more detaild explanation?

[OSuwaidi]

I do agree that example 2.23 seems to be lacking details and could be confusing at first.

In this example, we're interested in understanding how the transformation matrix $A$, which is defined with respect to the canonical (standard) basis $[(1,0),(0,1)]$ , would be represented with respect to a new basis $B$. Essentially, we want to see how the same transformation $A$ behaves when we view everything from the perspective of the $B$ basis.

We can't simply apply $A$ on a coordinate vector with respect to $B$, because that is viewing it from the lens of the standard basis. So that would be transforming the coordinate vector, rather than the vector itself, which isn't our goal.

To correctly represent the transformation in the $B$-basis, we first convert the coordinates from the $B$-basis to the standard basis by applying $B$. Once in the standard basis, we can apply the transformation matrix defined on it: $A$, yielding $AB$. Finally, we express the result of the transformed vector in terms of $B$ (convert back to $B$-basis), so we take the inverse to get: $B^{-1}AB$, which gives you the answer $[(3,0),(0,1)]$.