Rooster

HC: woensdag, 15:15-17:00, Ruppert 134
WG: vrijdag, 11:00-12:45, Ruppert 135, op 30/9 BG 115

Tentamen en tussentoets: open boek.

Bijdragen tot het eindcijfer:

• Huiswerkopgaven: 15%
• Tussentoets: 35%
• Tentamen: 50%

ANNOUNCEMENTS

• Belangrijk: Degenen die de tussentoets niet hebben gehaald kunnen wel het eindtentamen doen, maar ze hebben geen recht op herkansing.
• Tussentoets van 2003: .pdf
• Tussentoets: woensdag, 12 oktober, 15:00-17:00, Ruppert, 134 (in plaats van HC). Material includes Part I of the program (see below).
• Reader: .pdf
• Geen WG vrijdag 14.10!!!
• Tussentoets: resultaten
• Tentamen: vrijdag, 11 november, 11-13 uur, Ruppert 135, open boek.
• Herkansing: 21 november, 14:00-16:00, BG 048.
• Proeftentamen van 2004 .pdf
• Korte uitwerking proeftentamen (Hado): .pdf
• Korte uitwerking regressie (Hado): .pdf
• Eindcijfers na herkansing:.xls

Lecture 1: Artificial neurons and networks. Activation functions, weights. Threshold linear activation function, a neuron computing binary "and" function.
Vectors, matrices, operations with them. Vector space Rⁿ. Geometric interpretation of vectors. Elementary transformations and Gaussian elimination method.

Lecture 2: Inner product of vectors and its properties. Geometric interpretation of solutions of a system of linear equations; lines in R² and hyperplanes in Rⁿ; normal vector to a hyperplane; linear subspaces of Rⁿ; linear independence; testing linear independence; linear span of a set of vectors; dimension of a vector space.

The following problem was considered at the workgroup.
Problem. Given two different points A and B in R² find an equation defining the straight line going through A and B.
Solution 1. The equation has the form ax+by=c. Substituting the coordinates of A for x and y validates the equation, similarly for B. So, we obtain a system of two linear equations with the unknowns a, b and c. Solve this system and take any non-zero solution for the sought a, b and c.
Solution 2. Vector (a,b) is orthogonal to the line defined by ax+by=c. (Why?) Also: (-b,a) is orthogonal to (a,b). Now let (u,v) be the coordinates of the vector B-A. Then we can take a=-v and b=u. How can we now determine c?

Problem for the WG: solve the same question in an n-dimentional space, that is, find the system of equations for the straight line going through two given points A and B. (It is easier to generalize Solution 2 above.)

More WG exercises: .pdf (Hado).

Lecture 3: Rank of a matrix; equality of column and row ranks; geometric interpretation of rank. Basis of a space; the standard basis in Rⁿ; coordinates of a vector in a given basis. The change of coordinates in Rⁿ; transfer matrix from one basis to another. Linear transformations, their matrices and geometric interpretation. Inverse matrix; regular matrices are invertible and vice versa; proof of the inversion algorithm by elementary transformations.

Lecture 4: Quadratic forms, their geometric interpretation. Matrix of a form. Transformation of matrices when coordinates are changed. Bringing a matrix to a canonical basis. Rank and index of a quadratic form.

Lecture 5: Derivatives of functions in one and several variables. Taylor series for functions in one variable. Partial derivatives. Gradient. Jacobi matrix.

Lecture 6: Total derivative. Derivatives of a composition of functions; chain formulas (K p. 444 + extra). Directional derivative. Gradient descent method. Second and higher order derivatives (K A59); Taylor formula up to second order. Local maxima and minima; investigation of extremum points (K p.990-993).

Lecture 7: Regression function. Linear regression. Least squares problem. Pseudoinverse matrix.

Part 1. Linear Algebra

• Vectors and matrices, matrix operations, Rⁿ  (K 6.1-6.2)
• Systems of linear equations and their matrix form; homogeneous and nonhomogeneous systems (K 6.3)
• Elementary transformations, bringing a matrix to a staircase (echelon) form (K 6.3)
• General solution of a system of linear equations; analysis of a staircase matrix; Gauss-Jordan elimination (K 6.3)
• Linear subspaces of Rⁿ; linear independence; linear span of a set of vectors; dimension of a space (K 6.4)
• Rank of a matrix; equality of column and row ranks; geometric interpretation of rank (K 6.4)
• Basis of a space; the standard basis in Rⁿ; coordinates of a vector in a given basis; changing a basis (K 6.4)
• Kronecker-Capelli theorem; fundamental system of solutions of a homogeneous system of equations (K 6.5)
• Inverse matrix; regular matrices are invertible and vice versa; inversion algorithm by elementary transformations (K 6.7 until Theorem 2)
• Linear transformations and  their matrices; linear and bilinear forms; quadratic forms (K p. 362-364,  p. 388)
• Bringing a quadratic form to a diagonal  matrix; positively and negatively definite forms (K p. 395-396)

• Inner product of vectors; metric (distance) and its basic properties (K p.361-362, 408-413)
• Balls in Rⁿ; open and closed sets; connected sets; vector functions; vector fields (K p. 423-424)
• Limits of (vector) functions in several variables; continuous functions and their properties (K p.425)
• Total derivative of a function; its geometric interpretation
• Partial derivatives; Jacobi matrix; gradient (K p. 425-427, 446-447, A57-A67)
• Directional derivatives; their geometric interpretation (K p.A57-A60)
• Derivatives of a composition of functions; chain formula (K p. 444 + extra)
• Second and higher order derivatives (K A59); Taylor formula
• Maxima and minima; search methods (K p.990-993)
• Regression function. Linear regression. Least squares problem. Pseudoinverse matrix.