Course synopsis Each module represents at least 12 hours (roughly 2 weeks' material)  

"Algebra"  "Analitica" 
Matrices and linear systems Matrices, sums and products Powers and inverses of square matrices Systems of equations, matrix form Elementary row operations Row reduction and rank Gauss(Jordan) method 
Vector analysis Vectors in space, operations on them Scalar product, distances, angles Cross product, triple product, areas, volumes Plane geometry, lines, conics Equation of a plane in space Parametric equation of a line 
Rank and dimension Solving a general system Row equivalence Subspaces of R^{n} Bases of subspaces Dimension of subspaces Spaces generated by rows and columns 
Differential calculus Curves in space, tangent vectors Arclength, line integrals Functions with domain in R^{n}, limits Continuity, topological notions Partial derivates, the gradient Vectorvalued functions, Jacobian matrix 
Linear mappings The notion of a vector space Bases, isomorphic vector spaces Linear mappings The matrix of a linear map Kernel and image, their dimensions The sum of two subspaces, Grassmann's formula 
Functions of two variables The graph of z=f(x,y), paraboloids Differentiability, tangent plane Second partial derivatives, Hessian matrix Taylor expansion of f(x,y) Critical points Extreme values 
Diagonalization Linear transformations, change of basis Eigenvalues, determinants Eigenvectors, eigenspaces, multiplicity Symmetric matrices, quadratic forms Orthogonal matrices and rotations Diagonalization of square matrices 
Surfaces and quadrics Surfaces of revolution, spheres Ellipsoids and hyperboloids Rotations in R^{n} Quadrics, planes and lines Surfaces defined implicitly Normal vector, tangent plane 
Provisional lecture plan L = 60 hours and E = 39 hours  

14/03  L  Matrices, sums and products [Summary]  Notes, pp 14  
15/03  L  Powers and inverses of square matrices [Summary]  Notes, pp 58  
17/03 


18/03  E  
18/03  E  Vectors in space, operations on them  Notes, pp 3334  
21/03  L  Systems of equations, matrix form [Summary]  Notes, pp 910  
22/03  L  Elementary row operations [Summary]  Notes, pp 1314  
24/03  L  Gauss(Jordan) method [Summary]  Notes, pp 1718  
25/03  E  Notes, pp 10,12,16  
25/03  E  Scalar product, distances, angles 
Notes, pp 3536  
28/03  L  Row reduction and rank [Summary]  Notes, p 1819  
29/03  L  Solving a general system [Summary]  Notes, pp 2122  
31/03  L  Row equivalence [Summary]  Notes, pp 11,15,20  
01/04  E  Notes, pp 2223  
01/04  E  Cross product, triple product, areas, volumes  Notes, pp 3940  
04/04  L  Subpsaces of R^{n} [Summary]  Notes, pp 2526  
05/04  L  Bases of subspaces [Summary]  Notes, pp 2930  
07/04  L  Dimension and rank [Summary]  Notes, pp 3132  
15/04  E  
15/04  E  Spaces generated by rows and columns  Notes, pp 2627  
11/04  L  The notion of a vector space [Summary]  Notes, pp 4950  
12/04  L  More bases and subspaces [Summary]  Notes, pp 5155  
14/04  L  Linear mappings [Summary]  
15/04  E  
15/04  E  Notes, pp 5051  
18/04  L  Bases and linear mappings [Summary]  Notes, pp 5759  
19/04  L  Linear maps, matrices, systems [Summary]  
EASTER  
28/04  L  Linear maps, kernels, images [Summary]  
02/05  L  The sum of two subspaces [Summary]  Notes, pp 6566  
03/05  L  Eigenvectors and eigenvalues [Summary]  Notes, pp 6971  
05/05  L  Eigenspaces [Summary]  Notes, pp 7778  
06/05  E  
06/05  E  Determinants  
09/05  L  Diagonalization of square matrices [Summary]  Notes, pp 8182  
10/05  L  Diagonalizing symmetric matrices [Summary]  Notes, pp 8586  
12/05  L  Orthogonal matrices and quadratic forms [Summary]  
13/05  E  
13/05  E  Plane geometry, conics  Notes, p 91 only  
16/05  L  Equation of a plane [Summary]  Notes, pp 4143  
17/05  L  Parametric equation of a line [Summary]  Notes, pp 4546  
19/05  L  Curves in space [Summary]  MAI, pp279283  
20/05  E  
20/05  E  
23/05  L  Line integrals along curves [Summary]  MAI, pp368375  
24/05  L  Functions to and from R^{n} [SUMMARY]  MAI, p76, p285  
26/05  L  Partial derivatives, the gradient  MAI, p286  
27/05  E  
27/05  E  Graphs z=f(x,y), paraboloids  
30/05  L  Chain rule for scalar functions [SUMMARY]  MAI, p288289  
31/05  L  Scalar functions of 2 and 3 variables [SUMMARY]  MAII, p156162  
03/06  E  
03/06  E  
06/06  L  Second order partial derivatives [SUMMARY]  MAII, p168169  
07/06  L  First Taylor expansions [SUMMARY]  MAII p161  
09/06  L  Critical points [SUMMARY]  MAII, p174180  
09/06  E  Extreme values (Room 1S at 4pm)  
10/06  E  
10/06  E  
13/06  L  Conics and quadrics [SUMMARY]  Notes, p 9395  
14/06  L  Quadrics, conics and lines [SUMMARY]  
16/06  L  Examples and interpretations (at 1000) [Summary]  
E  Model exam solutions (at 1130)  
17/06  E  
17/06  E  
20/06  L  Parametrization of surfaces [Summary]  
21/06  L  Surfaces of revolution [Summary]  
23/06  L  Tangents and normals (at 10.00) [Summary corrected]  
E  Worked sample exercises 