1. The problem statement, all variables and given/known data I have a vector V with components v1, v2in some basis and I want to switch to a new (orthonormal) basis a,b whose components in the old basis are given. I want to find the representation of vector V in the new orthonormal basis i.e. find the components va,vb such that |v⟩ = va|a⟩+vb|b⟩. 2. Relevant equations Original vector |v⟩ = (1+i)1 (√3+i)2 Vector |a⟩ = [(1+i√3)/4]1, [(-√3(1+i))/√8]2 Vector |b⟩ = [(√3(1+i))/√8]1[(i+√3)/4]2 new vector |v⟩ = va|a⟩+vb|b⟩. Where the subscripts denote the row number. 3. The attempt at a solution I took the inner product between the given vectors |a⟩ and |v⟩ for va and |b⟩ and |v⟩ for vb: va = ⟨a|v⟩ and vb=⟨b|v⟩ I don't think this is right - because for this to be true the new vector v using these components would have a norm which is invariant under the transformation and it does not.