Find basis of orthogonal complement of space W I am supposed to find a basis of orthogonal complement of space W = $[(1,1,0,1),(0,1,0,1), (0,0,0,1)]$ .
I have already found an orthogonal basis of W, which is $[(1,1,0,1), (-2/3,1/3,0,1/3) (0,-1/2,0,1/2)]$ 
How would I find the basis of orthogonal complement? 
I don't know how to begin, however my textbook says, it is supposed to be an easy problem, basically without any calculations. Thanks!
 A: Well, the vector space $W$ is a 3-dim.  subspace of ${\Bbb R}^4$ and so the orthogonal complement $W^\perp$ is 1-dimensional. Looking at the basis of $W$, it is clear that $W^\perp$ is generated by $(0,0,1,0)$.
A: The orthogonal completement of $W$ is the set of all vectors that are each orthogonal to all of the vectors in $W$.
The basis of the orthogonal complement, then, is a set of vectors such that all of the vectors in the orthogonal complement can be written as a linear combination of these vectors.
Let’s first take a simpler example in $\mathbb{R}^3$. If we have a subspace with basis $\{(1,0,0),(0,1,0)\}$ then this is a two-dimensional subspace—it describes a plane in 3D. It should be fairly obvious that there can only be one more dimension within which we can have vectors that are orthogonal to all of the vectors in our subspace, and therefore the basis will have only one vector. Can you come up with a single vector that is perpendicular to all the vectors in our subspace? $(0,0,1)$ works, and this is indeed the basis of the orthogonal complement.
Can you answer your question now?
