Abstract
This chapter is devoted mainly to Euclidean vector spaces and their transformations. It starts with notions of inner product, length, angle, Gramian, orthogonality, orthonormal basis, etc. Orthogonal transformations are investigated, and orientation of Euclidean spaces is discussed. After that, symmetric linear transformations of real vector spaces are investigated in greater detail; for instance, a basic property of such transformations to have a real eigenvalue and real eigenvector is provided with three proofs, based on different principles, and the extremal properties of the eigenvalues of a symmetric linear transformation are discussed (the Courant–Fischer theorem). Theoretical results and notions are accompanied with numerous examples from different areas, including mechanics and geometry; examples include elements of the theory of hypersurfaces in a Euclidean space (principal curvatures, Euler’s formula, etc.). At the end of the chapter, pseudo-Euclidean vector spaces and Lorentz transformations (analogue of orthogonal transformations for pseudo-Euclidean spaces) are considered in greater detail.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Infinite-dimensional Euclidean spaces are usually called pre-Hilbert spaces. An especially important role in a number of branches of mathematics and physics is played by the so-called Hilbert spaces, which are pre-Hilbert spaces that have the additional property of completeness, just for the case of infinite dimension. (Sometimes, in the definition of pre-Hilbert space, the condition (x,x)>0 is replaced by the weaker condition (x,x)≥0.)
- 2.
The molecules of amino acids likewise determine a certain orientation of space. In biology, the two possible orientations are designated by D (right = dexter in Latin) and L (left = laevus). For some unknown reason, they all determine the same orientation, namely the counterclockwise one.
- 3.
It was published as L.B. Nisnevich, V.I. Bryzgalov, “On a problem of n-dimensional geometry,” Uspekhi Mat. Nauk 8:4(56) (1953), 169–172.
- 4.
This example is taken from Gantmacher and Krein’s book Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, Moscow 1950, English translation, AMS Chelsea Publishing, 2002.
- 5.
The more customary point of view, when the hypersurface (for example, a curve or surface) consists of points, requires the consideration of an n-dimensional space consisting of points (otherwise affine space), which will be introduced in the following chapter.
- 6.
For example, the circle of radius 1 with center at the origin with Cartesian coordinates x,y can be defined not only by the formula x=cost, y=sint, but also by the formula x=cosτ, y=−sinτ (with the replacement t=−τ), or by the formula x=sinτ, y=cosτ (replacement \(t= \frac{\pi}{2}-\tau\)).
- 7.
Examples of surfaces consisting entirely of elliptic points are ellipsoids, hyperboloids of two sheets, and elliptic paraboloids, while surfaces consisting entirely of hyperbolic points include hyperboloids of one sheet and hyperbolic paraboloids.
- 8.
We remark that this terminology differs from what is generally used: Our “spacelike” vectors are usually called “timelike,” and conversely. The difference is explained by the condition s=n−1 that we have assumed. In the conventional definition of Minkowski space, one usually considers the quadratic form −x 2−y 2−z 2+t 2, with index of inertia s=1, and we need to multiply it by −1 in order that the condition s≥n/2 be satisfied.
- 9.
For example, a Lorentz transformation of Minkowski space—a four-dimensional pseudo-Euclidean space—plays the same role in the special theory of relativity (which is where the term Lorentz transformation comes from) as that played by the Galilean transformations, which describe the passage from one inertial reference frame to another in classical Newtonian mechanics.
- 10.
The nondegeneracy of the subspace L 0=(0) relative to a bilinear form follows from the definitions given on pages 266 and 195. Indeed, the rank of the restriction of the bilinear form to the subspace (0) is zero, and therefore, it coincides with dim(0).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Shafarevich, I.R., Remizov, A.O. (2012). Euclidean Spaces. In: Linear Algebra and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30994-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-30994-6_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30993-9
Online ISBN: 978-3-642-30994-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)