Abstract
In this chapter, we shall introduce and discuss some necessary basic notions, which mainly belong to Analytic Geometry and Linear Algebra. We assume that the reader is already familiar with these disciplines, so our introduction will be brief.
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Notes
- 1.
The elements of linear spaces are conventionally called “vectors”. In the subsequent sections of this chapter, we will introduce another definition of vectors and also definition of tensors. It is important to remember that in the sense of this definition, all vectors and tensors are elements of some linear spaces and, therefore, are all “vectors”. For this reason, starting from the next section, we will avoid using the word “vector” when it means an element of arbitrary linear space.
- 2.
In case where the matrix elements are allowed to be complex, the matrix that satisfies the property \(U^\dagger =U^{-1}\) is called unitary. The operation \(U^\dagger \) is called Hermitian conjugation and consists of complex conjugation plus transposition \(U^\dagger =(U^*)^T\).
- 3.
One can prove that it is always possible to replace the rotation of the rigid body around an arbitrary axis by performing the sequence of particular rotations \(\,\hat{\wedge }_{z}(\alpha )\, \hat{\wedge }_{y}(\beta )\,\hat{\wedge }_{z}(-\alpha )\,\,\). The proof can be done using the Euler angles (see, e.g., [1, 2]).
- 4.
In the second part of the book, we shall see that this is the Lorentz transformation in special relativity in the units \(c=1\).
References
L.D. Landau, E.M. Lifshits, Mechanics—Course of Theoretical Physics, vol. 1 (Butterworth-Heinemann, 1976)
J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley Pub Co., 1994)
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Shapiro, I.L. (2019). Linear Spaces, Vectors, and Tensors. In: A Primer in Tensor Analysis and Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-26895-4_1
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DOI: https://doi.org/10.1007/978-3-030-26895-4_1
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