I don't like:

> Changing coordinates. Given a vector x expressed in the standard basis, we change coordinates by applying

    x_eig = V^−1 x.

> This operation does not change the vector itself; it only changes how the vector is represented, converting it into eigenvector coordinates.

I'm sure the author knows about the difference between the vector and a coordinate vector given a basis. I just don't like this wording. It's weird to say applying a function doesn't affect a change.