Are All Bases Sets of Orthogonal Vectors?

In summary, orthogonormal vectors are a set of vectors that are both perpendicular to each other and have a magnitude of 1. They are important because they form a basis for vector spaces, making calculations and transformations easier. Orthogonal vectors are simply at right angles to each other, while orthonormal vectors are a subset of orthogonal vectors with a magnitude of 1. To find basis vectors, the Gram-Schmidt process can be used. These basis vectors are closely related to coordinate systems and can be used to represent any point in the system as a linear combination of the basis vectors.
  • #1
pr0me7heu2
14
2
Would all bases be sets of orthogonal (but not necessarily orthonormal) vectors?
 
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  • #2
No. Try to think of a basis for R^2 consisting of nonorthogonal vectors -- there are plenty.
 
  • #3
For example, {[itex]\vec{i}[/itex], [itex]\vec{i}+ \vec{j}[/itex] } is a basis for R2 and they are not orthogonal (with the "usual" inner product). It happens to be easier to to find components in an orthonormal basis.

In any case, "orthogonal" as well as "orthonormal" depend upon an innerproduct defined on the vector space. Given any basis it is always possible to define an innerproduct in which that basis is orthonormal.
 

FAQ: Are All Bases Sets of Orthogonal Vectors?

What are orthogonormal vectors?

Orthogonormal vectors are a set of vectors that are both orthogonal (perpendicular) and normalized (have a magnitude of 1). In other words, they are at right angles to each other and have a length of 1.

Why are orthogonormal vectors important?

Orthogonormal vectors are important because they form a basis for vector spaces. This means that any vector in the space can be represented as a linear combination of the orthogonormal vectors. This makes calculations and transformations in vector spaces much easier.

What is the difference between orthogonal and orthonormal vectors?

Orthogonal vectors are simply at right angles to each other, while orthonormal vectors are not only perpendicular but also have a magnitude of 1. In other words, orthonormal vectors are a subset of orthogonal vectors.

How do you find basis vectors?

To find basis vectors, you can use the Gram-Schmidt process, which is a mathematical algorithm for finding an orthogonal (or orthonormal) basis for a vector space. This involves taking a set of linearly independent vectors and transforming them into a set of orthogonal (or orthonormal) vectors.

What is the relationship between basis vectors and coordinate systems?

Basis vectors are closely related to coordinate systems. In fact, the basis vectors of a coordinate system are the unit vectors that point in the direction of each coordinate axis. By using these basis vectors, any point in the coordinate system can be uniquely represented as a linear combination of the basis vectors, also known as coordinates.

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