28.17. Orthogonal Transformations

Try these first

1. 1.

   What condition must a matrix meet to qualify as an orthogonal transformation?

   Hint

   Consider the identity matrix.

2. 2.

   Name one property preserved by orthogonal transformations.

   Hint

   Think about what happens to vectors during rotation.

3. 3.

   What is the orthogonal transformation condition for a matrix?

   • AT A = 0
   • AT A = I
   • AT A = A

   Hint

   Think about identity matrices.

4. 4.

   True or False: Orthogonal transformations distort the lengths of vectors.

   • True
   • False

   Hint

   Recall the definition of orthogonality.

5. 5.

   Given a 2D vector x = (3, 4), find the image of this vector under a 90-degree rotation orthogonal transformation.

   Hint

   Set up the transformation matrix for the specified rotation.

6. 6.

   Explain why an orthogonal transformation might be preferred in modeling physical systems in engineering.

   Hint

   Think about what happens to structure representations under various transformations.