1) Using the indentity:||u+v||^2=||u||^2+||v||^2+(u,v)+(v,u) - where (x,y) denotes the inner product . Now if we let the second term = iv, my book gives the identity as then: ||u+iv||^2=||u||^2+||v||^2-i(u,v)+i(v,u) . And I am struggling to derive this myself, here is my working: ||u+iv||^2=(u,u)+(u,iv)+(iv,u)+(iv,iv) = ||u||^2+(u,iv)+i(v,u)+||iv||^2 Now the two terms I am struggling with are why ||iv||^2=||v||^2 and trying to get the (u,iv) to -i(u,v) as it is in . My attempt was to use the hermiticity property to give (u,iv)=(iv,u)* - letting * denote the complex conugate - gives (u,iv)=(-iv,u)=-i(v,u), where the last equivalence follws from linearity in the first factor, rather than -i(u,v). 2) Is to find an orthornomal basis for the orthogonal complement of the subspace spanned by (2,1-i,0,1) and (1,0,i,3), under the standard inner product. Now I am ok with the main concepts of the strategy needed here: the orthogonal complement of the subspace is given by the space of solutions to 2a+(1-i)b+d=0 and, a+ic+3d=0, obtained by ensuring that any arbitarty vector that lies in this space (a,b,c,d) is orthogonal to the spanning vectors given of the subspace. HOWEVER, my book gives these equations (1) and (2) as 2a+(1+i)b+d=0 and a-i+3d=0, and I can not see why the 'i's have been multipled by -1 compared to my workng. Many Thanks for any assistance.