Question

Select 3 vectors that are linear independent than adjust the base to be orthogonal

<1,1,0> <0,0,1> <-1,2,0> <-1,2,1>

show ALL POSSIBLE work evein betweennbatween

Answer 1

Let v₁ = (1,1,0), v₂ = (0,0,1), v₃= (-1,2,0) and v₄= (-1,2,1). Apparently, since v₄ = v₂+v₃, and since none of v₁,v₂,v₃ can be a scalar multiple or a linear combination of the other 2 vectors, hence { v₁,v₂,v₃ } is a linearly independent set.

Now, let u₁=v₁=(1,1,0), u₂=v₂-proj_u1(v₂)= v₂-[(v₂.u₁)/(u₁.u₁)]u₁ = v₂-[(0+0+0)/(1+1+0)u₁ = v₂= (0,0,1), and u₃ = v₃-proj_u1(v₃)-proj_u2(v₃)=v₃-[(v₃.u₁)/(u₁.u₁)]u₁-[(v₃.u₂)/(u₂.u₂)]u₂=v₃-[(-1+2+0)/(1+1+0)]u₁-[(0+0+0)/(0+0+1]u₂= (-1,2,0) –(1/2)(1,1,0)= (-3/2,3/2,0).Then {u₁,u₂,u₃ }={(1,1,0),(0,0,1), (-3/2,3/2,0)}is an orthogonal set.