Orthogonal Transformations An orthogonal transformation is a linear transformation that preserves the inner product of vectors. In simpler terms, it rotates...
Orthogonal Transformations
An orthogonal transformation is a linear transformation that preserves the inner product of vectors. In simpler terms, it rotates and scales vectors in such a way that the resulting vectors are aligned along the principle axis, resulting in a change of angle but preserving the length of the vectors.
Examples:
Rotation: A clockwise rotation by 90 degrees is an orthogonal transformation.
Scaling: Multiplying all components of a vector by a constant is an orthogonal transformation.
Reflection: Reflecting a vector across the y-axis is an orthogonal transformation.
Properties of Orthogonal Transformations:
Orthogonal transformations are linear transformations.
The kernel of an orthogonal transformation is a subspace of the domain.
The range of an orthogonal transformation is a subspace of the codomain.
Orthogonal transformations are invertible if and only if the linear transformation is one-to-one and onto.
Applications of Orthogonal Transformations:
Orthogonal transformations have numerous applications in various fields, including:
Signal processing: Orthogonal transformations are used to remove noise and interference from signals.
Data analysis: Orthogonal transformations can be used to reduce dimensionality and identify patterns in data sets.
Physics: Orthogonal transformations are used in quantum mechanics to describe the behavior of particles.
Computer graphics: Orthogonal transformations are used to create realistic images and animations