Syllabus
*Affine space
- Affine space, subspace.
- Coordinates and their transformation.
- Linear combination of points. Definition of basic geometrical figures in plane, segment of line and its center, triangle, center of gravity.
- Parametric equations of subspace.
- (n-1)-dimensional subspace and its equation.
- Subspace as intersection of (n-1)-dimensional subspaces.
*Euclidean space
- Vector spaces with scalar product, geometrical interpretation of scalar product.
- Outer and vector product, geometrical interpretation. Axioms of measure.
- Euclidean space and subspace, equation of (n-1)-dimensional subspace.
- Cartesian coordinates.
- Orthogonal subspaces.
- Distance from a point to a subspace, distance of two subspaces.
- Angle and its measure, angle of a line and a subspace.
*Set of points satisfying a given property
- Set of points defined by distance; axis of a segment of line, angle, belt.
- Circle of Apollonios; power of a circle with respect to the point, chordal of two circles, chordal center of three circles.
- General equation of a conic section, classification, singular and regular conic sections. Equations of regular conic sections and their properties. Conic sections as sections of a cone.
Annotation
Analytical geometry of affine and Euclidean spaces and their subspaces. Sets of points defined by distance.
This subject provides the high-school analytical geometry with theoretical base using linear algebra.