. Use Gram-Schmidt process to transform the basis (1, 1, 0), (0, 1, 1) and (1, 0, 1) into an orthonormal basis for
R3
in the same order.
To transform the given basis (1, 1, 0), (0, 1, 1), and (1, 0, 1) into an orthonormal basis for R3 using the Gram-Schmidt process, follow these steps:
Step 1: Normalize the first vector
Normalize the first vector (1, 1, 0) by dividing it by its magnitude:
v1 = (1, 1, 0)
||v1|| = sqrt(1^2 + 1^2 + 0^2) = sqrt(2)
v1_normalized = (1/sqrt(2), 1/sqrt(2), 0)
Step 2: Calculate the projection of the second vector onto the first vector
To find the projection of the second vector (0, 1, 1) onto the first vector (1/sqrt(2), 1/sqrt(2), 0), use the formula:
proj(v2, v1) = (v2 · v1_normalized) * v1_normalized
where · represents the dot product.
v2 = (0, 1, 1)
v1_normalized = (1/sqrt(2), 1/sqrt(2), 0)
proj(v2, v1) = ((0, 1, 1) · (1/sqrt(2), 1/sqrt(2), 0)) * (1/sqrt(2), 1/sqrt(2), 0)
= (1/sqrt(2), 1/sqrt(2), 0) * (1/sqrt(2), 1/sqrt(2), 0)
= (1/2, 1/2, 0)
Step 3: Subtract the projection from the second vector
Subtract the projection (1/2, 1/2, 0) from the second vector (0, 1, 1) to obtain a new vector orthogonal to the first vector:
v2_orthogonal = v2 - proj(v2, v1)
= (0, 1, 1) - (1/2, 1/2, 0)
= (-1/2, 1/2, 1)
Step 4: Normalize the orthogonal vector
Normalize the orthogonal vector (-1/2, 1/2, 1) by dividing it by its magnitude:
v2_normalized = (-1/2, 1/2, 1) / ||(-1/2, 1/2, 1)||
= (-1/2, 1/2, 1) / sqrt((-1/2)^2 + (1/2)^2 + 1^2)
= (-1/2, 1/2, 1) / sqrt(1/2 + 1/2 + 1)
= (-1/2, 1/2, 1) / sqrt(3/2)
= (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
Step 5: Calculate the projection of the third vector onto the first vector and orthogonal vector
To find the projection of the third vector (1, 0, 1) onto the first vector (1/sqrt(2), 1/sqrt(2), 0) and the orthogonal vector (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2), use the same formula as Step 2.
proj(v3, v1_orthogonal) = (v3 · v1_orthogonal) * v1_orthogonal
proj(v3, v2_normalized) = (v3 · v2_normalized) * v2_normalized
v3 = (1, 0, 1)
v1_orthogonal = (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
v2_normalized = (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
proj(v3, v1_orthogonal) = ((1, 0, 1) · (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)) * (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
= (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2) * (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
= (-1/2, 1/2, sqrt(6)/2)
proj(v3, v2_normalized) = ((1, 0, 1) · (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)) * (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
= (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2) * (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
= (-1/2, 1/2, sqrt(6)/2)
Step 6: Subtract the projections from the third vector
Subtract the projection (-1/2, 1/2, sqrt(6)/2) from the third vector (1, 0, 1) to obtain a new vector orthogonal to both the first and orthogonal vectors:
v3_orthogonal = v3 - proj(v3, v1_orthogonal) - proj(v3, v2_normalized)
= (1, 0, 1) - (-1/2, 1/2, sqrt(6)/2) - (-1/2, 1/2, sqrt(6)/2)
= (1, 0, 1) + (1/2, -1/2, -sqrt(6)/2) + (1/2, -1/2, -sqrt(6)/2)
= (2, -1, -sqrt(6))
Step 7: Normalize the orthogonal vector
Normalize the orthogonal vector (2, -1, -sqrt(6)) by dividing it by its magnitude:
v3_normalized = (2, -1, -sqrt(6)) / ||(2, -1, -sqrt(6))||
= (2, -1, -sqrt(6)) / sqrt(2^2 + (-1)^2 + (-sqrt(6))^2)
= (2, -1, -sqrt(6)) / sqrt(4 + 1 + 6)
= (2, -1, -sqrt(6)) / sqrt(11)
= (2/sqrt(11), -1/sqrt(11), -sqrt(6)/sqrt(11))
Therefore, the orthonormal basis in the same order is:
v1_normalized = (1/sqrt(2), 1/sqrt(2), 0)
v2_normalized = (-sqrt(2)/2, sqrt(2)/2, sqrt(6)/2)
v3_normalized = (2/sqrt(11), -1/sqrt(11), -sqrt(6)/sqrt(11))