- #1
Nusc
- 760
- 2
What is the difference between these terms?
In what context do they apply to?
How important is it that we treat them differently?
In what context do they apply to?
How important is it that we treat them differently?
When speaking of two vectors u,v perpendicular and orthoganal are used interchangably to mean that an inner product is zero.Nusc said:What is the difference between these terms?
In what context do they apply to?
How important is it that we treat them differently?
Orthonormal, orthogonal, and perpendicular are often used interchangeably, but they have different meanings. Orthonormal refers to a set of vectors or functions that are both orthogonal and normalized. Orthogonal means that two vectors are perpendicular to each other, or at a right angle. Perpendicular specifically refers to two lines or vectors that intersect at a 90 degree angle.
Orthonormal vectors are useful in many areas of mathematics and science, particularly in linear algebra, signal processing, and quantum mechanics. In linear algebra, orthonormal vectors can be used as a basis for vector spaces, making calculations and proofs simpler. In signal processing, they can be used to decompose signals into simpler components. In quantum mechanics, they are used to describe the state of a quantum system.
A matrix is orthonormal if its columns (or rows) form an orthonormal set of vectors. This means that the columns (or rows) are both orthogonal and normalized. In other words, the dot product of any two columns (or rows) is 0, and the length of each column (or row) is 1. Orthonormal matrices are important in linear transformations and can be used to preserve the length and angle between vectors.
Yes, a set of vectors can be orthonormal in one basis but not in another. This is because the concept of orthogonality depends on the inner product used. Different inner products can lead to different definitions of orthogonality. Therefore, a set of vectors that may be orthogonal in one basis may not be orthogonal in another basis with a different inner product.
To determine if a set of vectors is orthonormal, you can use the following steps: