Orthogonal Projection Problems?

In summary: Just try to do this directly and ignore any fancy words. Can you write down an expression for arbitrary vector in the span of 1 and##e^t##? Then try to compute the norm.
  • #1
ashah99
60
2
Thread moved from the technical forums and poster has been reminded to show their work
Summary:: Hello all, I am hoping for guidance on these linear algebra problems.
For the first one, I'm having issues starting...does the orthogonality principle apply here?
For the second one, is the intent to find v such that v(transpose)u = 0? So, could v = [3, 1, 0](transpose) work?

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  • #2
For the second I think you have a correct answer (there are many choices). For the first, what is the orthogonality principle?
 
  • #3
Office_Shredder said:
For the second I think you have a correct answer (there are many choices). For the first, what is the orthogonality principle?
The way it was explained to me was that given a vector x in a sub space, find a closest point x_hat that is in S. Not sure if that’s the right approach for the first problem.
And yes I think you’re right on the second part on numerous answers, I wanted confirmation so thanks for that. Another answer I can think of is [4 0 -1]^T.
 
  • #4
ashah99 said:
The way it was explained to me was that given a vector x in a sub space, find a closest point x_hat that is in S. Not sure if that’s the right approach for the first problem.

That certainly sounds like what you're trying to do. Why don't you try it out and post your computation here?
 
  • #5
Office_Shredder said:
That certainly sounds like what you're trying to do. Why don't you try it out and post your computation here?
Not sure about the formula to use. Have any suggestions?
 
  • #6
ashah99 said:
Not sure about the formula to use. Have any suggestions?

Just try to do this directly and ignore any fancy words. Can you write down an expression for arbitrary vector in the span of 1 and##e^t##? Then try to compute the norm. You should get a quadratic formula in two unknown variables.
 

1. What is an orthogonal projection problem?

An orthogonal projection problem is a mathematical problem that involves finding the closest point from a given set of points to a given line or plane. This is achieved by projecting the point onto the line or plane at a right angle, which is known as an orthogonal projection.

2. What are some real-life applications of orthogonal projection problems?

Orthogonal projection problems have many applications in various fields such as engineering, computer graphics, and physics. For example, in engineering, these problems are used in structural analysis to determine the forces acting on a structure. In computer graphics, they are used to create 3D images by projecting 2D images onto a 3D plane. In physics, they are used to analyze the motion of objects in space.

3. How do you solve an orthogonal projection problem?

To solve an orthogonal projection problem, you first need to identify the given set of points and the line or plane onto which the points will be projected. Then, you can use the formula for orthogonal projection to find the closest point to the given set of points. This involves finding the perpendicular distance from the point to the line or plane and using it to calculate the coordinates of the projected point.

4. What is the difference between orthogonal projection and oblique projection?

The main difference between orthogonal projection and oblique projection is the angle at which the projection is made. In orthogonal projection, the projection is made at a right angle, while in oblique projection, the projection is made at an angle other than 90 degrees. This results in a distorted image in oblique projection, whereas orthogonal projection preserves the shape and size of the original object.

5. Are there any limitations to using orthogonal projection?

While orthogonal projection is a useful tool in many applications, it does have some limitations. One limitation is that it only works for points that are close to the line or plane onto which they are projected. If the points are far from the line or plane, the projection may not accurately represent the original points. Additionally, orthogonal projection assumes that the line or plane is fixed, which may not always be the case in real-world scenarios.

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