Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Functions
- 2 Permutations of a finite set
- 3 Groups of permutations of ℝ and ℂ
- 4 The Möbius group
- 5 The regular solids
- 6 Abstract groups
- 7 Inversions of the Möbius plane and stereographic projection
- 8 Equivalence relations
- 9 Cosets
- 10 Direct product
- 11 Fields and vector spaces
- 12 Linear transformations
- 13 The general linear group GL(2, F)
- 14 The vector space V3(F)
- 15 Eigenvectors and eigenvalues
- 16 Homomorphisms
- 17 Conjugacy
- 18 Linear fractional groups
- 19 Quaternions and rotations
- 20 Affine groups
- 21 Orthogonal groups
- 22 Discrete groups fixing a line
- 23 Wallpaper groups
- Bibliography
- Index
19 - Quaternions and rotations
Published online by Cambridge University Press: 05 June 2012
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Functions
- 2 Permutations of a finite set
- 3 Groups of permutations of ℝ and ℂ
- 4 The Möbius group
- 5 The regular solids
- 6 Abstract groups
- 7 Inversions of the Möbius plane and stereographic projection
- 8 Equivalence relations
- 9 Cosets
- 10 Direct product
- 11 Fields and vector spaces
- 12 Linear transformations
- 13 The general linear group GL(2, F)
- 14 The vector space V3(F)
- 15 Eigenvectors and eigenvalues
- 16 Homomorphisms
- 17 Conjugacy
- 18 Linear fractional groups
- 19 Quaternions and rotations
- 20 Affine groups
- 21 Orthogonal groups
- 22 Discrete groups fixing a line
- 23 Wallpaper groups
- Bibliography
- Index
Summary
In this chapter we first extend the algebra of matrices by defining matrix addition and establishing the distributive laws. We then use matrices to define the quaternions and develop some of their special algebraic properties. Finally we show how the group of inner automorphisms of the quaternions is isomorphic to the group of rotations of 3-dimensional space with a given fixed point.
Concurrent reading: Birkoff and MacLane, chapter 8, section 10; Rees, pp. 42–44; Curtis, chapter 5; Coxeter (1974), chapter 6.
Addition of matrices
In the first two questions we develop the algebra of matrices in a quite general context. We define addition on matrices of the same shape in a way which coincides with componentwise addition, and we show that this addition is compatible with our definition of matrix multiplication in the sense that both distributive laws hold.
1 If v ↦ vA and v ↦ vB are both linear transformations Vn(F) → Vm(F), prove that v ↦ vA + vB is a linear transformation.
This establishes the existence of a matrix C such that vA + vB = vC for all vectors v. We now define A + B = C and this operation is called addition of matrices.
By considering v = (1, 0, …, 0), (0, 1, …, 0), etc., show that C is formed from A and B by componentwise addition.
2 […]
Information
- Type
- Chapter
- Information
- GroupsA Path to Geometry, pp. 178 - 184Publisher: Cambridge University PressPrint publication year: 1985
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the PDF of this book is currently unknown
and may be updated in the future.
- 3
- Cited by