Find the minimal distance from the point a to the space of a

Find the minimal distance from the point a to the space of all linear combinations...

Find the minimal distance from the point a = [1 -1 -1 2] to the space of all linear combinations of the vectors e_1 = [1 -1 0 0], e_2 = [1 0 -1 0] and e_3 = [1 0 0 -1].

Solution

We first apply Gram-Schmidt orthogomalization process to the vectors e₁, e₂, e₃, a. The desired distance will be |v₄|

Let v₁ = e₁ = (1,-1,0,0)^t

Now v₂ = e₂ - <e₂, v₁>/<e₁,e₁> . e₁

=> v₂ = (1,0,-1,0)^t - 1/2(1,-1,0,0)^t = (-1/2, 1/2, -1,0)^t

v₃ = e₃ - <e₃,v₁>/<e₁,e₁>.e₁ - <e₃, v₂>/<e₂, e₂> .e₂

=> v₃ = (1,0,0,-1)^t - 1/2 (1,-1,0,0)^t +1/4 (1,0,-1,0)^t = (3/4,1/2,-1/4,-1)^t

v₄ = a - <a,v₁>/<e₁, e₁> .e₁ - <a,v₂>/<e₂, e₂> . e₂ - <a,v₃>/<e₃, e₃> .e₃

=> v₄ = (1,-1,-1,2)^t - (1,-1,0,0)^t - 0(1,0,-1,0)^t +3/4(1,0,0,-1)^t

=> v₄ = (3/4, 0, -1, 5/4)^t

|v₄| = sqrt(9/16 + 0 + 1 + 25/16) = sqrt(34/16 + 1) = sqrt(50/16) = 5sqrt(2)/4