## Affine and metric geometry based on linear algebra |

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### Contents

AFFINE GEOMETRY | 1 |

Intersection of Affine Subspaces ll | 11 |

Parallelism | 19 |

25 other sections not shown

### Common terms and phrases

a e 0(V a e k affine space affine subspace affine transformation anisotropic space assume automorphism C+(V Clifford Algebra commutative completely determined complex numbers Consequently Consider contains coordinate system defined denote dimensional division ring elements Euclidean plane Euclidean space example exceptional plane Exercise finite fixed point ft(V given hence homomorphism hyperbolic pair hyperbolic plane hyperbolic space hyperplane im(a involution isometry isomorphism isotropic lines k-algebra ker(a kernel Lemma linear subspace linear transformation linearly independent Lorentz plane magnification mapping metric vector space Moreover non-singular plane non-singular space non-zero isotropic vector non-zero vector null space one-to-one orthogonal basis orthogonal group parallel pointwise fixed Proof Proposition Prove real numbers reflection rigid motion rotation group scalar semi-affine transformation singular space of dimension square subgroup subset Suppose Sylvester invariant symmetric bilinear function symmetric matrix three-dimensional translation unique Witt Theorem