Find the formula for the projection P:C^3--->C^3 on W along U, where:

W=span{(1,i,0),(0,i,1)} and

U=span{(0,1,0)}.

Attempt at solution:

According to my textbook: A function T: V -> V is called the projection on W1 along W2 if, for x = x1 + x2 with x1 E W1 and x2 E W2 we have T(x)=x1, but this requires that V be the direct sum of W1 and W2. For the above question W is taken over the real field F=R so for W to be a subspace of C^3, C^3 must be taken over the field R but then C^3 has dimension 6 (example of basis: {(1,0,0),(0,1,0),(0,0,1),(i,0,0),(0,i,0),(0,0,i)}) . Since the vectors in span W and span U are independent they form a basis for W and U respectively and although the intersection of the 2 basis is the empty set the sum of the dimensions is 3 and not 6 so C^3 is not the direct sum of W and U. Or do we just assume that it is the direct sum then x1= a(1,i,0)+b(0,i,1) and x2=c(0,1,0) and x=x1+x2=(a,ai+bi+c,b) and T(x)=T(a,ai+bi+c,b)=(a,ai+bi,b) where a,b,c are scalars and since T=T^2 this is a projection.

This is the only definition the textbook gives for a projection so I'm really uncertain about how to approach the question and what to do. Any help would be greatly appreciated.Thanks in advance.