Chapter 1.- Bilinear maps and quadratics forms. Bilinear maps, bilinear forms, quadratic forms and real quadratic forms.

Chapter 2.- Homogenous tensors and duality. Duaility, homogenous tensor, operations with homogeneous tensors. Symmetry and skewsymmetry.

PART 2.- Euclidean vector spaces.

Chapter 1.- Introduction to euclidean spaces. Scalar product. Norm of a vector. Properties. Angle between two vectors.

Chapter 2.- Orthogonality. Ortoghonal vectors, orthogonal systems. Gram-Schmidt methos. Singularities of orthonormal bases. Orthogonal projections. Symmetric endomorphisms.

Chapter 3.- Orthogonal maps. Definition and properties, eigenvalues and eigenvectors. Orientation of a basis. Inverse and direct orthogonal maps. Classification on orthogonal maps in two and three dimensions .

Chapter 4.- Cross product and mixed product. Definition and properties.

PART 3.- Affine geometry.

Chapter 1.- Affine space.  Definition and properties. Systems of reference. Affine varieties. Pencils of affine varieties. Angle and distance between affine varieties. Affine transformations.

Chapter 2.- Projective space. Introduction. Homogenenous coordinates. Proper points and points at infinity. Reference change in homogneous coordinates. Equations of affine varieties in homogeneous coordinates. 

Chapter 1.- Conic curves. Definition and equations. Intersection of a line and a conic. Polarity. Important points and lines of a conic. Description of nondegenerated conics: ellipse, hyperbola and parabola. Change of reference. Classification of conics and reduced equations. Pencil of conics.

Chapter 2.- Quadric surfaces. Definition and equations. Intersection of a line and a quadric. Polarity. Change of reference.  Important points and lines of a quadric. Classification of quadrics and reduced equations. Description of the quadric surfaces of rank 3 and 4.