Calculus Orthonormal

Question

Find the orthogonal projection of

V = 0, 0, 3, 0

onto the subspace V of R4 spanned by [1, -1, -1, 1], [-1, -1, -1, -1], [-1, -1, 1,1]

Answer 1

(-3/4, 3/4 , 9/4, 3,4)

To find the orthogonal projection of V = (0, 0, 3, 0) onto the subspace W of R^4 spanned by the vectors [1, -1, -1, 1], [-1, -1, -1, -1], and [-1, -1, 1, 1], follow these steps:

1. Normalize the basis vectors.
2. For [1, -1, -1, 1], the magnitude is sqrt(1^2 + (-1)^2 + (-1)^2 + 1^2) = 2.
3. Normalized vector: (1/2, -1/2, -1/2, 1/2).
4. For [-1, -1, -1, -1], the magnitude is sqrt((-1)^2 + (-1)^2 + (-1)^2 + (-1)^2) = 2.
5. Normalized vector: (-1/2, -1/2, -1/2, -1/2).
6. For [-1, -1, 1, 1], the magnitude is sqrt((-1)^2 + (-1)^2 + 1^2 + 1^2) = 2.
7. Normalized vector: (-1/2, -1/2, 1/2, 1/2).
8. Compute the projection of V onto each normalized basis vector.
9. For the first vector (1/2, -1/2, -1/2, 1/2):
10. Dot product = (0)(1/2) + (0)(-1/2) + (3)(-1/2) + (0)(1/2) = -3/2.
11. Projection = (-3/2) * (1/2, -1/2, -1/2, 1/2) = (-3/4, 3/4, 3/4, -3/4).
12. For the second vector (-1/2, -1/2, -1/2, -1/2):
13. Dot product = (0)(-1/2) + (0)(-1/2) + (3)(-1/2) + (0)(-1/2) = -3/2.
14. Projection = (-3/2) * (-1/2, -1/2, -1/2, -1/2) = (3/4, 3/4, 3/4, 3/4).
15. For the third vector (-1/2, -1/2, 1/2, 1/2):
16. Dot product = (0)(-1/2) + (0)(-1/2) + (3)(1/2) + (0)(1/2) = 3/2.
17. Projection = (3/2) * (-1/2, -1/2, 1/2, 1/2) = (-3/4, -3/4, 3/4, 3/4).
18. Add up the projections to get the total projection onto the subspace W.
19. (-3/4, 3/4, 3/4, -3/4) + (3/4, 3/4, 3/4, 3/4) + (-3/4, -3/4, 3/4, 3/4) = (-3/4 + 3/4 - 3/4, 3/4 + 3/4 - 3/4, 3/4 + 3/4 + 3/4, -3/4 + 3/4 + 3/4).
20. Simplify: (-3/4, 3/4, 9/4, 3/4).

Final result:

The orthogonal projection of V = (0, 0, 3, 0) onto the subspace W is (-3/4, 3/4, 9/4, 3/4).