Orthogonal projection and reflection (matrices)

In summary, orthogonal projection and reflection are mathematical operations using matrices. They have various uses in science, including computer graphics, signal processing, and physics. These operations are represented by projection and reflection matrices, constructed using basis vectors. The main difference between them is that projection projects a vector onto a subspace, while reflection reflects it across a line or plane. In linear algebra, they are used for solving equations, computing eigenvalues and eigenvectors, and other matrix operations, with applications in computer science, economics, and statistics.
  • #1
Clandry
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Homework Statement


[Imgur](http://i.imgur.com/VFT1haQ.png)

Homework Equations


reflection matrix = 2*projection matrix - Identity matrix

The Attempt at a Solution


Using the above equation, I get that B is the projection matrix and E is the reflection matrix.
Can someone please verify if this is indeed the correct procedure?
 
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  • #2
Your justification for the projection matrix might need some fleshing out.
The equation for the reflection matrix justifies your choice of E if you can show that B is a projection matrix.
 

1. What is orthogonal projection and reflection?

Orthogonal projection and reflection are mathematical operations performed using matrices. Orthogonal projection involves projecting a vector onto a subspace, while reflection involves reflecting a vector across a line or plane.

2. What are the uses of orthogonal projection and reflection in science?

Orthogonal projection and reflection have a wide range of uses in science, including computer graphics, signal processing, and physics. In computer graphics, they are used to create 3D images and animations. In signal processing, they are used to remove noise from signals. In physics, they are used to model the behavior of light and other waves.

3. How are orthogonal projection and reflection represented using matrices?

Orthogonal projection and reflection are represented using square matrices called projection matrices and reflection matrices, respectively. These matrices are constructed using the basis vectors of the subspace or line/plane onto which the vector is being projected or reflected.

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

The main difference between orthogonal projection and reflection is that projection involves projecting a vector onto a subspace, while reflection involves reflecting a vector across a line or plane. This means that the result of projection is a vector in the subspace, while the result of reflection is a vector that is symmetric about the line/plane of reflection.

5. How are orthogonal projection and reflection used in linear algebra?

Orthogonal projection and reflection are fundamental concepts in linear algebra and are used extensively in various applications. They are used to solve systems of linear equations, compute eigenvalues and eigenvectors, and perform other operations on matrices. They also have applications in fields such as computer science, economics, and statistics.

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