Is the decomposition of a vector dependent on the inner product space?
currently taking linear algebra and i an dealing with inner product spaces. specifically, dealing with orthogonal and orthonormal basis. for the grant shmit process, I understand everything expect literally the first step. the first step is decomposing the vector.
what i understand: a standard basis, let's say R^2, is a basis that is orthonormal. any vector within the space can we decomposed into its corresponding x and y position using rcos(theta) and rsin(theta) respectively.
but, how does this work if the basis isn't:
(1) unit orthogonal
and
(2) standard
additionally, does the does the first step of the grant process have decomposition, and if it does am I thinking of it properly?
I am not looking for anything formal at all.
please try and keep it simple if you can.
thank you very much!