(a) An orthonormal basis for R^3 using the Gram-Schmidt orthogonalization procedure for set B1 is: ((1/√2, 0, 1/√2), (1/√6, 2/√6, 1/√6), (-1/√3, 2/√3, -1/√3)).
(b) An orthonormal basis for R^3 using the Gram-Schmidt orthogonalization procedure for set B2 is: ((2/√6, 1/√6, 1/√6), (1/√6, -1/√6, √2/√6), (-1/√17, 1/√17, 2/√17)).
(a) Applying the Gram-Schmidt orthogonalization procedure to set B1 = {(1,0,1),(1,1,0),(1,1,2)}:
Step 1: Normalize the first vector:
v1 = (1,0,1)
u1 = v1 / ||v1|| = (1,0,1) / √(1^2 + 0^2 + 1^2) = (1,0,1) / √2 = (√2/2, 0, √2/2)
Step 2: Compute the projection of the second vector onto the subspace spanned by u1:
v2 = (1,1,0)
proj = (v2 · u1) / (u1 · u1) * u1 = ((1,1,0) · (√2/2, 0, √2/2)) / ((√2/2, 0, √2/2) · (√2/2, 0, √2/2)) * (√2/2, 0, √2/2)
= (√2/2) / (1/2 + 1/2) * (√2/2, 0, √2/2) = (√2/2) * (√2/2, 0, √2/2) = (1/2, 0, 1/2)
Step 3: Orthogonalize v2 by subtracting the projection:
u2 = v2 - proj = (1,1,0) - (1/2, 0, 1/2) = (1/2, 1, -1/2)
Step 4: Normalize u2:
u2 = u2 / ||u2|| = (1/2, 1, -1/2) / √(1/4 + 1 + 1/4) = (1/2, 1, -1/2) / √2 = (1/√8, √2/√8, -1/√8) = (1/√8, √2/4, -1/√8)
Step 5: Compute the projection of the third vector onto the subspace spanned by u1 and u2:
v3 = (1,1,2)
proj1 = (v3 · u1) / (u1 · u1) * u1 = ((1,1,2) · (√2/2, 0, √2/2)) / ((√2/2, 0, √2/2) · (√2/2, 0, √2/2)) * (√2/2, 0, √2/2)
= (√2) / (1/2 + 1/2) * (√2/2, 0, √2/2) = (√2) * (√2/2, 0, √2/2) = (1, 0, 1)
proj2 = (v3 · u2) / (u2 · u2) * u2 = ((1,1,2) · (1/√8, √2/4, -1/√8)) / ((1/√8, √2/4, -1/√8) · (1/√8, √2/4, -1/√8))
= (√2) / (1/8 + 2/8 + 1/8) * (1/√8, √2/4, -1/√8) = (√2) * (1/√8, √2/4, -1/√8) = (1, √2/2, -1)
proj = proj1 + proj2 = (1, 0, 1) + (1, √2/2, -1) = (2, √2/2, 0)
Step 6: Orthogonalize v3 by subtracting the projection:
u3 = v3 - proj = (1,1,2) - (2, √2/2, 0) = (-1, 1 - √2/2, 2)
Step 7: Normalize u3:
u3 = u3 / ||u3|| = (-1, 1 - √2/2, 2) / √((-1)^2 + (1 - √2/2)^2 + 2^2) = (-1, 1 - √2/2, 2) / √(3 - 2√2 + 2 + 4) = (-1, 1 - √2/2, 2) / √(9 - 2√2) = (-1/√(9 - 2√2), (1 - √2/2)/√(9 - 2√2), 2/√(9 - 2√2))
Therefore, an orthonormal basis for R3 using the Gram-Schmidt orthogonalization procedure for set B1 is:
u1 = (√2/2, 0, √2/2)
u2 = (1/√8, √2/4, -1/√8)
u3 = (-1/√(9 - 2√2), (1 - √2/2)/√(9 - 2√2), 2/√(9 - 2√2))
(b) Applying the Gram-Schmidt orthogonalization procedure to set B2 = {(2,1,1),(1,0,1),(0,0,2)}:
Step 1: Normalize the first vector:
v1 = (2,1,1)
u1 = v1 / ||v1|| = (2,1,1) / √(2^2 + 1^2 + 1^2) = (2,1,1) / √6 = (2/√6, 1/√6, 1/√6)
Step 2: Compute the projection of the second vector onto the subspace spanned by u1:
v2 = (1,0,1)
proj = (v2 · u1) / (u1 · u1) * u1 = ((1,0,1) · (2/√6, 1/√6, 1/√6)) / ((2/√6, 1/√6, 1/√6) · (2/√6, 1/√6, 1/√6)) * (2/√6, 1/√6, 1/√6)
= (√6/3) / (2/3 + 1/6 + 1/6) * (2/√6, 1/√6, 1/√6) = (√6/3) * (2/√6, 1/√6, 1/√6) = (2/3, 1/3, 1/3)
Step 3: Orthogonalize v2 by subtracting the projection:
u2 = v2 - proj = (1,0,1) - (2/3, 1/3, 1/3) = (1/3, -1/3, 2/3)
Step 4: Normalize u2:
u2 = u2 / ||u2|| = (1/3, -1/3, 2/3) / √((1/3)^2 + (-1/3)^2 + (2/3)^2) = (1/3, -1/3, 2/3) / √(1/9 + 1/9 + 4/9) = (1/3, -1/3, 2/3) / √(6/9) = (1/√6, -1/√6, 2/√6) = (1/√6, -1/√6, √2/√6)
Step 5: Compute the projection of the third vector onto the subspace spanned by u1 and u2:
v3 = (0,0,2)
proj1 = (v3 · u1) / (u1 · u1) * u1 = ((0,0,2) · (2/√6, 1/√6, 1/√6)) / ((2/√6, 1/√6, 1/√6) · (2/√6, 1/√6, 1/√6)) * (2/√6, 1/√6, 1/√6)
= (2√6/3) / (2/3 + 1/6 + 1/6) * (2/√6, 1/√6, 1/√6) = (2√6/3) * (2/√6, 1/√6, 1/√6) = (4/3, 2/3, 2/3)
proj2 = (v3 · u2) / (u2 · u2) * u2 = ((0,0,2) · (1/√6, -1/√6, √2/√6)) / ((1/√6, -1/√6, √2/√6) · (1/√6, -1/√6, √2/√6))
= (2√2/3) / (1/6 + 1/6 + 2/6) * (1/√6, -1/√6, √2/√6) = (2√2/3) * (1/√6, -1/√6, √2/√6) = (√2/3, -√2/3, 2/3√2)
proj = proj1 + proj2 = (4/3, 2/3, 2/3) + (√2/3, -√2/3, 2/3√2) = (4/3 + √2/3, 2/3 - √2/3, 2/3 + 2/3√2) = ((4 + √2)/3, (2 - √2)/3, (2 + 2√2)/3)
Step 6: Orthogonalize v3 by subtracting the projection:
u3 = v3 - proj = (0,0,2) - ((4 + √2)/3, (2 - √2)/3, (2 + 2√2)/3) = (-4/3 - √2/3, -2/3 + √2/3, 2/3 - 2/3√2)
Step 7: Normalize u3:
u3 = u3 / ||u3|| = (-4/3 - √2/3, -2/3 + √2/3, 2/3 - 2/3√2) / √((-4/3 - √2/3)^2 + (-2/3 + √2/3)^2 + (2/3 - 2/3√2)^2)
= (-4/3 - √2/3, -2/3 + √2/3, 2/3 - 2/3√2) / √(16/9 + 8/9 - 8√2/9 + 8/9 + 4/9 + 8√2/9 + 4/9 - 8/9 + 8/9)
= (-4/3 - √2/3, -2/3 + √2/3, 2/3 - 2/3√2) / √(36/9 + 16/9 + 16/9)
= (-4/3 - √2/3, -2/3 + √2/3, 2/3 - 2/3√2) / √(68/9)
= (-√2/√68, √2/√68, 2√2/√68)
= (-1/√17, 1/√17, 2/√17)
Therefore, an orthonormal basis for R3 using the Gram-Schmidt orthogonalization procedure for set B2 is:
u1 = (2/√6, 1/√6, 1/√6)
u2 = (1/√6, -1/√6, √2/√6)
u3 = (-1/√17, 1/√17, 2/√17)
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