Linear Algebra: Orthogonal Projections

[triucsd]

Hey first time poster here.

Homework Statement

Find the orthogonal projection projWy

u1 = [-1; 3; 1; 1], u2 = [3; 1; 1; -1], u3 = [-1; -1; 3; -1], y = [1; 0; 0; 1]

where {u1, u2, u3} is an orthogonal basis.

Homework Equations

yhat = [dot(y,U1)/dot(U1,U1)]U1 + ... [dot(y,Un)/dot(Un,Un)]Un

The Attempt at a Solution

I used the equation above and I ended up getting:

0u1 + 1/6[3; 1; 1;-1] + 1/6[-1; -1; 3; -1] = 1/6[2; 0; 4; -2]

the answer given by my professor is 1/6[-1; -1; 3; -1] and I don't understand why. I can get that answer by not adding dot(y,U3)/dot(U3,U3)]U3 to yhat, but I suspect that isn't the right way to do it. Where am I going wrong?

 

[mighty2000]

Dear first time poster,

Thank you for reaching out to us for help with your problem. It seems like you have a good understanding of the concept of orthogonal projection, but there may be a small error in your calculation.

To find the orthogonal projection of a vector onto a subspace spanned by an orthogonal basis, we can use the formula you have provided: yhat = [dot(y,U1)/dot(U1,U1)]U1 + ... [dot(y,Un)/dot(Un,Un)]Un. However, in this formula, the dot product of the vector with itself (e.g. dot(U1,U1)) should result in a scalar, not a vector. This may be where you are going wrong.

Let's go through the calculation step by step to see where the issue may lie. First, we can find the dot products of y with each of the basis vectors:

dot(y,u1) = -1
dot(y,u2) = 0
dot(y,u3) = -1

Next, we can find the dot products of each basis vector with itself:

dot(u1,u1) = 12
dot(u2,u2) = 12
dot(u3,u3) = 12

Now, we can plug these values into the formula to find the orthogonal projection:

yhat = [(-1)/12]u1 + [(0)/12]u2 + [(-1)/12]u3
= [-1/12][-1; 3; 1; 1] + [0][3; 1; 1; -1] + [-1/12][-1; -1; 3; -1]
= [1/12; -1/4; -1/12; -1/12] + [0; 0; 0; 0] + [1/12; 1/12; -1/4; 1/12]
= [2/12; -1/4; -1/6; -1/12]

As you can see, this result is slightly different from the one you obtained. It is possible that you may have made a small error in your calculation of the dot products, or in adding the terms together. I would suggest double-checking your work to see where the discrepancy may lie.

I hope this helps clarify the issue for you. If you have any further questions, please