Teaching GuideTerm 2021/222020/212019/202018/192017/182016/172015/162014/152013/142012/132011/122010/112008/092009/10 Higher Technical University College of Civil Engineering
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 Grao en Tecnoloxía da Enxeñaría Civil Subjects Álxebra lineal II Contents
 Topic Sub-topic Bilinear maps and homogenous tensors. 1. Bilinear maps and quadratic forms. 1.1 Bilinear maps. 1.2 Bilinear forms. 1.3 Quadratic forms. 1.4 Real quadratic forms. 2. Homogenous tensors and duality. 2.1 Duality. 2.2 Homogenous tensor. 2.3 Operations with homogenous tensors. 2.4 Simmetry and skewsimmetry. Euclidean vectorial spaces. 1. Introduction to euclidean spaces. 1.1 Scalar product. 1.2 Norm of a vector. Properties. 1.3 Angle between two vectors. 2. Orthogonality. 2.1 Orthogonal vectors. 2.2 Orthogonal systems. Gram-Schmidt method. 2.3 Singularties of orthonormal basis. 2.4 Orthogonal projection. 2.5 Symmetric endomorphisms. 3. Orthogonal maps. 3.1 Definition. 3.2 Properties. 3.3 Eigenvalues and eigenvectors of an orthogonal map. 3.4 Orientation of a basis 3.5 Inverse and direct orthogonal maps. 3.6 Classiication of orthogonal maps in two and three dimensions. 4. Vectorial product and triple product. 4.1 Definition. 4.2 Properties. Affine geometry. 1. Affine space. 1.1 Definition and properties. 1.2 System of reference. 1.3 Affine varieties. 1.4 Pencils of affine varietes. 1.5 Distances and angles between affine varieties. 1.6 Affine transformations. 2. Projective space. 2.1 Introduction. 2.2 Homogeneous coordinates. 2.3 Proper points and points at infinity. 2.4 Reference change in homogeneous coordinates. 2.5 Equations of affine varieties in homogeneous coordinates. Conics and quadric surfaces. 1. Conics. 1.1 Definition and equations. 1.2 Intersections of a conic and a line. 1.3 Polarity. 1.4 Important potins and lines of a conic. 1.5 Description of nondegenerated conics: ellipse, parabola e hyperbola. 1.6 Change of reference. 1.7 Classification of conics. Reduced equation. 1.8. Pencils of conics. 2. Quadric surfaces. 2.1 Definition and equations. 2.2 Intersections of a quadric surface and a line. 2.3 Polarity. 2.4 Change of reference. 2.5 Important potins, lines and planes of a quadric surface. 2.6 Classification of quadric surfaces. Reduced equation. 2.7 Description of quadric surfaces of rank 3 and 4.
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