Please answer with explanation and clear writting Let W spa

Please answer with explanation and clear writting

Let W = span{(1, 1, 1, 1), (3, 1, 3, 1), (6, 2, 4, 0)}. Find an orthonormal basis of W using the Gram-Schmidt process. (a) {(1/2, 1/2, 1/2, 1/2), {(1/2, -1/2, 1/2, -1/2), {(1/2, 1/2, -1/2, -1/2)} (b) {(1/2, 1/2, 1/2, 1/2), {(0, -1/2, 1, -1/2), {(1/squareroot 6, 0, 1/squareroot 6, -1/squareroot 6)} (c) {(1/2, 1/2, 1/2, 1/2), {(0, 1/squareroot 2, -1/squareroot 2, 0), {(1/squareroot 2, 0, 0, -1/squareroot 2)} (d) {(1/2, 1/2, 1/2, 1/2), {(1/squareroot 2, -1/squareroot 2, 0, 0), {(0, 0, 1/squareroot 2, -1/squareroot 2)} (e) {(1/2, 1/2, 1/2, 1/2), {(-1/squareroot 2, 0, 1/squareroot 2, 0), {(0, 1/squareroot 2, 0, 1/squareroot 2)}

Solution

Let v₁ = (1,1,1,1), v ₂= (3,1,3,1),and v₃ = (6,2,4,0). These vectors are linearly independent as the RREf of the matrix with these vectors as columns is

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Now, let u₁ = v₁ = (1,1,1,1), u₂ = v₂ –proj_u1(v₂) = v₂-[(v₂.u₁)/(u₁.u₁)]u₁ = v₂-[(3+1+3+1)/(1+1+1+1)]u₁ = (3,1,3,1)- 2(1,1,1,1)= (3,1,3,1)- (2,2,2,2)= (1,-1,1,-1).

Also, u₃ = v₃ –proj_u1(v₃) - proj_u2(v₃) = v₃ -[(v₃.u₁)/(u₁.u₁)]u₁-[(v₃.u₂)/(u₂.u₂)]u₂ = v₃ -[(6+2+4+0)/(1+1+1+1)]u₁       -[(6-2+4-0)/(1+1+1+1)]u₂ = (6,2,4,0) – 3(1,1,1,1)-2(1,-1,1,-1) = (6,2,4,0) –(3,3,3,3)-(2,-2,2,-2) = (1,1,-1,-1)

Then {u₁,u₂,u₃} is an orthogonal basis for W. We have to only convert these into unit vectors. Let w₁ = u₁/|| u₁|| = (1,1,1,1)/(1+1+1+1) = 1/2(1,1,1,1) = (1/2,1/2,1/2,1/2). Also, w₂= u₂/|| u₂|| = 1/2(1,-1,1,-1)= (1/2,-1/2,1/2,-1/2) and w₃ = u₃/|| u₃||= 1/2(1,1,-1,-1) = (1/2,1/2,-1/2,-1/2). Then {w₁,w₂,w₃} is an orthonormal basis for W. Op[tion (a) is the correct answer.

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