Question

Transform the basis {[1, 0, 1], [0, 1, 2], [2, 1, 0]} for R^3 into an orthonormal basis, using the Gram-Schmidt process.

Answer 1

Define proj(u, v) to be the projection of u onto v. proj(u, v) = v*(u dot v)/(v dot v)

||u|| = norm u

From the 3 given vectors, we want to form a basis such that each basis vector is orthogonal to every other and an unit vector.

Take e1 = ||[1,0,1]|| = (1/sqrt(2))[1,0,1]. The first basis vector is arbitrary.

e2 = ||[0,1,2] - proj([0,1,2], [1,0,1])||. We remove the non-orthogonal part to ensure e2 is orthogonal. Then take the norm to make it an unit vector.

e3 = ||[2,1,0] - proj([2,1,0], [0,1,2]) - proj([2,1,0], [1,0,1])||. As with e2, remove every part orthogonal to e1 or e2 to ensure orthogonality. Then take the norm to make it an unit vector.