Expanding a given vector into another orthonormal basis

In summary, the conversation discusses equations 9.2.25 and 9.2.32, which define the inner product of two vectors in terms of their components in the same basis. The question is raised regarding the calculation of ## \langle 1| V \rangle ## and whether or not the given vectors are assumed to be in an orthogonal basis. The author is assumed to be somewhat sloppy in their writing, leading to confusion.
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Pushoam
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Equation 9.2.25 defines the inner product of two vectors in terms of their components in the same basis.
In equation 9.2.32, the basis of ## |V \rangle## is not given.
## |1 \rangle ## and ## |2 \rangle ## themselves form basis vectors. Then how can one calculate ## \langle 1| V \rangle ## ?
Do we have to assume that all of these vectors are given in the orthogonal basis with vectors given by
## \begin{bmatrix}
1
\\ 0
\end{bmatrix} ## and ## \begin{bmatrix}
0
\\ 1
\end{bmatrix} ## ?
 

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  • #2
The author is assuming that the components of all of the vectors in 9.2.32 and 9.2.33 are with respect to the same basis. They are being somewhat sloppy in their writing, which is probably the source of your confusion.

jason
 
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1. What is the purpose of expanding a given vector into another orthonormal basis?

The purpose of expanding a given vector into another orthonormal basis is to provide a new set of coordinates for the vector that is easier to work with. This new basis can simplify calculations and make it easier to understand the relationships between different vectors.

2. How is a vector expanded into an orthonormal basis?

To expand a vector into an orthonormal basis, we use a process called Gram-Schmidt orthogonalization. This involves finding a set of orthogonal vectors that span the same subspace as the original vector, and then normalizing these vectors to create an orthonormal basis.

3. What is the difference between an orthogonal and an orthonormal basis?

An orthogonal basis is a set of vectors that are all perpendicular to each other. An orthonormal basis is a set of vectors that are not only orthogonal, but also have a magnitude of 1. This means that in an orthonormal basis, all vectors are perpendicular and have a length of 1.

4. Can any vector be expanded into an orthonormal basis?

Yes, any vector can be expanded into an orthonormal basis. This is because we can always find a set of orthogonal vectors that span the same subspace as the original vector, and then normalize them to create an orthonormal basis.

5. What are some applications of expanding a vector into an orthonormal basis?

Expanding a vector into an orthonormal basis has many applications in mathematics, physics, and engineering. It is commonly used in linear algebra, signal processing, and quantum mechanics, among others. It can help simplify calculations, solve equations, and understand the relationships between different vectors in a given space.

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