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Independence of the sum of any number of vectors on brackets
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abscissa axis afﬁne coordinate system afﬁne plane afﬁne space afﬁne transformation algebra axioms basis e1 bivector called centre of symmetry circle coefﬁcients collinear complex numbers conjugate Coor deﬁned Deﬁnition Desargues theorem direction vector elementary transformations ellipse equal equation equivalent Euclidean plane Euclidean space expressed in terms family of vectors ﬁnd ﬁrst ﬁxed points follows formula given hence homogeneous hyperbola hyperboloid ideal points ideal straight line imaginary intersecting isomorphism Lemma linear fractional transformation linearly expressible linearly independent lines in space multiplication nates nonsingular nonzero notation obtain orientation orthogonal orthogonal matrix orthonormal basis parabola parallel pencil point in common point M0 preceding lecture projective plane projective-afﬁne plane Proof Proposition proves real-complex rectangular coordinates relation rotation satisﬁed scalar second-degree curve straight line passing sufﬁcient symbol system of rectangular theorem tion triple trivector uniquely determined vector space vectors e1 zero