An orthonormal system is a set of vectors that are both orthogonal (perpendicular to each other) and normalized (each vector has a length of 1). This means that the inner product of any two distinct vectors in the system is 0, and the inner product of any vector with itself is 1. An orthonormal basis is a specific type of orthonormal system that spans the entire vector space. This means that any vector in the space can be expressed as a linear combination of the basis vectors, and the basis vectors are linearly independent. In other words, an orthonormal basis is a complete set of vectors that can be used to represent any vector in the space. **

One can show an orthonormal basis by demonstrating that the basis vectors are orthogonal to each other and have a magnitude of 1. To show that the vectors are orthogonal, one can calculate the dot product of each pair of basis vectors and verify that the result is zero. Additionally, one can show that the magnitude of each basis vector is 1 by calculating the norm of each vector and confirming that it equals 1. Finally, one can demonstrate that the basis vectors span the entire vector space, ensuring that they form a complete basis. **

One can find an orthonormal basis by starting with a set of linearly independent vectors and then applying the Gram-Schmidt process. This process involves taking the original vectors and orthogonalizing them by subtracting off their projection onto the previously orthogonalized vectors. After orthogonalizing the vectors, they can be normalized to obtain an orthonormal basis. Another method is to use the eigenvalue decomposition of a symmetric matrix to find the orthonormal eigenvectors, which form an orthonormal basis for the matrix. **

The determinants of an orthonormal basis are the vectors that make up the basis. An orthonormal basis is a set of vectors that are both orthogonal (perpendicular to each other) and normalized (have a length of 1). The determinants of an orthonormal basis are the vectors that satisfy these two conditions. In other words, the determinants are the vectors that form the basis and allow for the representation of any vector in the space. **

An orthonormal basis in vector calculus is a set of vectors that are orthogonal (perpendicular) to each other and have a magnitude of 1. This means that the vectors are independent and form a basis for the vector space. Orthonormal bases are commonly used in vector calculus to simplify calculations and represent vectors in a clear and concise manner. By using an orthonormal basis, vector components can be easily determined and manipulated in various mathematical operations. **

Yes, the Hilbert space does not have an orthonormal Hamel basis. This is because a Hamel basis is a maximal linearly independent set, but it is not necessarily orthogonal. On the other hand, an orthonormal basis is a set of vectors that are not only linearly independent, but also orthogonal and normalized. Therefore, the two concepts are different, and the Hilbert space does not have an orthonormal Hamel basis. **

To extend a vector to an orthonormal basis, one can first normalize the given vector to make it a unit vector. Then, one can find a set of orthogonal vectors that are linearly independent from the given vector. Finally, one can normalize these orthogonal vectors to make them unit vectors, thus creating an orthonormal basis that includes the original vector. **

The order of vectors in an orthonormal basis does not matter because each vector in the basis is orthogonal to all the other vectors and has a unit length. This means that the vectors are linearly independent and span the same subspace regardless of their order. Therefore, the order in which the vectors are arranged does not affect the properties of the basis or the space it spans. **