Orthonormal Bases: Determining Coefficients for Arbitrary Vector

In summary, the conversation discusses the use of an orthonormal basis to express a vector in terms of a finite list of basis elements. This method, commonly used in results such as Gram-Schmidt, involves finding the coefficients of a vector by taking the inner product with each basis element. The geometric explanation for this approach is not clear, but it works because it operates on basis vectors. This is demonstrated by defining a linear operator and showing that it results in the original vector.
  • #1
mrxtothaz
14
0
If we have a vector that can be expressed in terms of some finite list of basis elements. If we have an orthonormal basis for a vector space V, then a vector v can be expressed as <v,e1>e1 +...+ <v,en>en. This appears to be widely used for many results (such as Gram-Schmidt), but the motivation for this is not clear to me. Not only that, I don't understand why this is the case (geometrically).

Obtaining the coefficients for a given vector (in terms of an orthonormal basis) using the inner product of the vector v with each basis element... why does this work?
 
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  • #2
If you consider the version of the dot product involving cosine, [itex]v\cdot e_{1}=|v||e_{1}|cos\theta=|v|cos\theta[/itex] because [itex]e_{1}[/itex] is a unit vector. This is just the "[itex]e_1[/itex] component" of v (rather than the x or y component of v).
I included a poorly done illustration in Paint.
 

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  • #3
It works because it works on basis vectors. Define the linear operator A by
[tex]Av = \sum_{i=1}^n \langle v, e_i \rangle e_i.[/tex]
Then for any k,
[tex]Ae_k = \sum_{i=1}^n \langle e_k, e_i \rangle e_i
= \sum_{i=1}^n \delta_{ki} e_i = e_k.[/tex]
It follows that A = I, so Av = v for any v.
 

Related to Orthonormal Bases: Determining Coefficients for Arbitrary Vector

1. What is an orthonormal basis?

An orthonormal basis is a set of vectors in a vector space that are mutually orthogonal (perpendicular) and have a unit length. This means that each vector in the basis is perpendicular to every other vector in the basis and has a magnitude of 1.

2. What is the purpose of determining coefficients for an arbitrary vector in an orthonormal basis?

The coefficients of an arbitrary vector in an orthonormal basis can be used to express the vector as a linear combination of the basis vectors. This allows for easier manipulation and calculation of the vector's properties.

3. How do you determine the coefficients for an arbitrary vector in an orthonormal basis?

The coefficients can be determined by taking the inner product (also known as the dot product) of the arbitrary vector with each basis vector. The result is the projection of the arbitrary vector onto the basis vector, which can then be used as the coefficient.

4. What is the significance of using an orthonormal basis in vector calculations?

Using an orthonormal basis allows for simpler and more efficient calculations because the basis vectors are perpendicular and have a unit length. This makes it easier to find the component of a vector in a specific direction and to perform operations such as vector addition and multiplication.

5. Can any set of vectors be used as an orthonormal basis?

No, not all sets of vectors can be an orthonormal basis. The vectors must be mutually orthogonal and have a unit length in order to qualify as an orthonormal basis. However, any set of vectors can be converted into an orthonormal basis through a process called Gram-Schmidt orthogonalization.

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