# Linear Algebra: Orthogonal Projections

• triucsd
In summary: Best of luck with your studies!In summary, to find the orthogonal projection of a vector onto a subspace spanned by an orthogonal basis, we can use the formula yhat = [dot(y,U1)/dot(U1,U1)]U1 + ... [dot(y,Un)/dot(Un,Un)]Un. However, it is important to note that the dot product of a vector with itself should result in a scalar, not a vector. By following this formula and calculating the dot products correctly, we can obtain the correct answer for the orthogonal projection.
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?

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

## What is a linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vector spaces and linear transformations. It involves the use of matrices, vectors, and systems of linear equations to solve problems and represent real-world situations.

## What are orthogonal projections?

Orthogonal projections are a method used in linear algebra to project a vector onto a subspace, while maintaining orthogonality. This means that the projected vector will be perpendicular to the subspace onto which it is being projected.

## How are orthogonal projections used in real life?

Orthogonal projections have many practical applications in fields such as engineering, physics, and computer graphics. They are used to model and solve problems involving linear systems, such as finding the best fit line for a set of data points or calculating forces in a mechanical system.

## What is the difference between orthogonal projections and regular projections?

The main difference between orthogonal projections and regular projections is that orthogonal projections preserve the original vector's length and direction, while regular projections do not necessarily maintain these properties. Additionally, in orthogonal projections, the projected vector will be perpendicular to the subspace, whereas regular projections do not have this requirement.

## How do you calculate an orthogonal projection?

To calculate an orthogonal projection, you can use the formula P = (v dot u / u dot u) * u, where P is the projected vector, v is the original vector, and u is a unit vector representing the subspace. This formula is based on the concept that the projection of v onto u is equal to the scalar projection of v onto u, multiplied by the unit vector u.

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