Whilst studying linear algebra I came accross an at first sight peculiar conclusion. The double basis representation of the Identity linear transformation equals the Identity matrix of same dimension as the basis in question.
Now my inquiry is if this pattern is generalisable. Say one were to take a set of basis vectors and represent them with respect to the basis from which they are an element of. Would this in turn lead to the set of standard basis vectors? Or is there perhaps a detail I am missing making this impossible?
To follow up based on the comment. The standard basis vectors which are the result, are these technically still within the coordinate system described by the initial set of basis vectors. Or are they within the standard coordinate system represented by the basis vector i, j, k?