Use the Gram-Schmidt process to transform the basis



1

1
1
,

0

1
1
,

2

4
3


for the Euclidean space R3 into an orthonormal basis for R3.

^ each of those 3 sets are vectors

see other post

http://www.jiskha.com/display.cgi?id=1428173198

Also, remember that
u3=u3-p(u1,w3)-p(u2,w3)

where p(u1,w3) is the projection of w3 on u1, etc.

To transform the given basis into an orthonormal basis using the Gram-Schmidt process, follow these steps:

Step 1: Start with the first vector in the basis: v1 = (1, 1, 0).

Step 2: Normalize v1 by dividing it by its length:
u1 = v1 / ||v1||,
where ||v1|| is the length of v1 given by the Euclidean norm:
||v1|| = sqrt(1^2 + 1^2 + 0^2) = sqrt(2).

Therefore, u1 = (1/sqrt(2), 1/sqrt(2), 0).

Step 3: Take the second vector in the original basis: v2 = (1, 1, 1).

Step 4: Subtract the projection of v2 onto u1 from v2:
w2 = v2 - (v2 · u1)u1,
where (v2 · u1) is the dot product of v2 and u1:
(v2 · u1) = (1)(1/sqrt(2)) + (1)(1/sqrt(2)) + (1)(0) = sqrt(2).

Therefore, w2 = (1, 1, 1) - sqrt(2)(1/sqrt(2), 1/sqrt(2), 0) = (1 - 1, 1 - 1, 1) = (0, 0, 1).

Step 5: Normalize w2 by dividing it by its length:
u2 = w2 / ||w2||,
where ||w2|| is the length of w2 given by the Euclidean norm:
||w2|| = sqrt(0^2 + 0^2 + 1^2) = 1.

Therefore, u2 = (0, 0, 1).

Step 6: Take the third vector in the original basis: v3 = (2, 4, 3).

Step 7: Subtract the projection of v3 onto u1 from v3:
w3 = v3 - (v3 · u1)u1,
where (v3 · u1) is the dot product of v3 and u1:
(v3 · u1) = (2)(1/sqrt(2)) + (4)(1/sqrt(2)) + (3)(0) = 3sqrt(2).

Therefore, w3 = (2, 4, 3) - 3sqrt(2)(1/sqrt(2), 1/sqrt(2), 0) = (2 - 3, 4 - 3, 3) = (-1, 1, 3).

Step 8: Subtract the projection of w3 onto u2 from w3:
w3 = w3 - (w3 · u2)u2,
where (w3 · u2) is the dot product of w3 and u2:
(w3 · u2) = (-1)(0) + (1)(0) + (3)(1) = 3.

Therefore, w3 = (-1, 1, 3) - 3(0, 0, 1) = (-1, 1, 3) - (0, 0, 3) = (-1, 1, 0).

Step 9: Normalize w3 by dividing it by its length:
u3 = w3 / ||w3||,
where ||w3|| is the length of w3 given by the Euclidean norm:
||w3|| = sqrt((-1)^2 + 1^2 + 0^2) = sqrt(2).

Therefore, u3 = (-1/sqrt(2), 1/sqrt(2), 0).

Step 10: The resulting orthonormal basis for R3 is:
{u1, u2, u3} = {(1/sqrt(2), 1/sqrt(2), 0), (0, 0, 1), (-1/sqrt(2), 1/sqrt(2), 0)}.