Question

let V=R^3 and S={u1,u2,u3}=[1;2;0],[1;0;0],[1;0;1] These are three vectors 1 by 3

use gschmidt to obtain an orthogonal basis and then find the coordinates of x=[1;2;3] relative to this basis.

Answer 1

recall the projection of b on a:
p_a(b) = a.b * a/(a.a)

v1 = u1 = (1,2,0)
v2 = u2 - p_v1(u2) = (1,0,0) - (1/5,2/5,0) = (4/5,-2/5,0)
v3 = u3 - p(u3,v1) - p(u3,v2) = (1,0,1) - (1/5,2/5,0) - (4/5,-2/5,0) = (0,0,1)

check:
v1.v2 = 4/5 - 4/5 = 0
v1.v3 = 0
v2.v3 = 0

so, v1,v2,v3 are orthogonal

Now you can find x in this system