I'm a bit confused about rotation and reflection matrices

Click For Summary
SUMMARY

The discussion centers on constructing rotation and reflection matrices in R3, specifically how to derive the matrix A that represents these transformations. To find A, one must analyze the standard basis vectors e_1, e_2, and e_3, determining their transformations under the given rotation or reflection. The transitioned vectors are then placed as columns in matrix A, which encapsulates the linear transformation. This method is a fundamental approach in linear algebra for representing transformations in three-dimensional space.

PREREQUISITES
  • Understanding of linear transformations in R3
  • Familiarity with matrix representation of transformations
  • Knowledge of standard basis vectors e_1, e_2, e_3
  • Basic concepts of rotation and reflection in geometry
NEXT STEPS
  • Study the derivation of rotation matrices in 3D space
  • Learn about reflection matrices and their properties
  • Explore the application of transformation matrices in computer graphics
  • Investigate the relationship between linear transformations and eigenvalues
USEFUL FOR

Students in linear algebra, mathematicians, computer graphics developers, and anyone interested in understanding geometric transformations in three-dimensional space.

PirateFan308
Messages
91
Reaction score
0

Homework Statement


I have a final coming up and I am a bit fuzzy on how to create a matrix that represents a rotation or reflection about a certain plane (in R3). Say we are given a rotation/reflection about either a plane or a line through two points T(v)=Av and we are told to find A. Do we simply have to look at e_1, e_2, e_3 and see what the transition would be for these vectors and plug the 'transitioned' e_1, e_2, e_3 into the columns of A, respectively? Or is there a lot more that I am missing? Thanks!
 
Physics news on Phys.org
Yup, that's how you find the columns of the matrix representing any linear transformation.
 

Similar threads

Graduate Questions about changing frame in QM and rotating wave approximation
  • · Replies 3 ·
Replies
3
Views
2K
Find matrix representation for rotating/reflecting hexagon
  • · Replies 2 ·
Replies
2
Views
4K
Graduate Phase difference between electric and magnetic dipole moment
  • · Replies 0 ·
Replies
0
Views
1K
Undergrad Factoring Matrices with Elementary Row Operations
  • · Replies 4 ·
Replies
4
Views
2K
MHB Matrices of Linear Transformations .... Poole, Example 6.76 ....
  • · Replies 2 ·
Replies
2
Views
1K
Graduate The meaning of ##(ict, x_1,x_2,x_3)##
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Index notation of vector rotation
  • · Replies 7 ·
Replies
7
Views
3K
Is My Matrix Calculation Correct for Composite Reflections in Linear Algebra?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Regarding transmission and reflection of EM-waves
  • · Replies 3 ·
Replies
3
Views
2K
MHB Symmetry Operations of a Cube: Geometric Descriptions and Matrix Representations
  • · Replies 31 ·
2
Replies
31
Views
3K