Is this a basis for the Bergman space?

Please help me find a basis for a subspace of a vector space.?

• Answer:

  The easy way to do this is to think about what this subspace actually is, geometrically. It's a plane through the origin that rises at about a 63-degree angle as you move "northeast." Its intersection with the x-y plane is along the line x+y=0 (i.e., the NW to SE line). So all we need is a pair of perpendicular unit vectors that lie in this plane. I would choose (-sqrt(1/2), +sqrt(1/2), 0) and (sqrt(1/6), sqrt(1/6), 2 sqrt(1/6)). Note that 1/6 + 1/6 + 4/6 = 1, so the second vector is a unit vector; the first vector is more obviously a unit vector. Their dot product is zero, showing that they are perpendicular to each other.