# Slides and detailed syllabus

AA 2015/16

The slides contain a weekly recap of the topics treated during the lectures, plus the exercises assignments. Later in the course, you won’t find explicit assignments: it is understood that you practice by doing the exercises at the end of each book section.

NB: the  slides are not exhaustive and should be used only  as a guideline for your individual study at home.

week1 week2  —  resolution of Ex 2 page 59 Matlab-Linsolve

Consistency of linear systems via ranks: rouché-capelli theorem

week3  Linear(in)dep (=week4) + exercises  ex-Oct6

Slides of week 5 and 6  week5&6 (new, with the invertibility theorem)

week7

Slides on  determinants & integrals

Slides on eigenvectors and eigenvalues, diagonalization:  week10&11 (new, with the Matlab exercise in extenso)

Exercises on  dyn_systems and solutions sol_dyn_systems

Detailed syllabus — page & Section numbering refers to the textbook

Sept 14, 2015: generalities on matrices and vectors. Scalar multiplication and addition. Examples and geometric interpretation of the operations in the case of vectors. Transposition and properties.  [PP 1-11]

Sept 17, 2015: linear and convex combinations. Examples (segment, average). Canonical vectors in R^n. Matrix-vector product: componentwise and as a linear combination of the columns. Examples: projections and reflections in R^2.  [PP 13-20]

Sept 22, 2015: Rotation in R^2 and matrix representation. Identity matrix I_n.  Properties of matrix-vector product. Linear equations. Hyperplane in R^n  and example of  its parametric representation in R^3. System of linear equations: generalities (solution set, inconsistency) and representation via matrix-vector product.  [PP 22-23-24, 27-28-29 and 31]

Sept 24, 2015:  Augmented matrix of a linear system. Row echelon form of a matrix. Resolution of a linear system in row echelon form. Parametric form of the solution set. [PP 29-35]

Sept 29, 2015: Gaussian Elimination. Resolution of a linear system via Gaussian Elimination. Rank and nullity of a matrix. Rouché-Capelli Theorem. [PP 41-49 and the file rouché-capelli]

Oct 1, 2015 : Solutions of a consistent system: degrees of freedom are nullA= n-rankA. Leontief model for an economy. Span of a set of vectors S in R^n. Examples. How to check wheter a vector b belongs to Span S. Generators of R^n.The dimension of an ambient space as the minimal number of generators. Subspaces: definition, examples and dimension. [Pag 56, beginnig of Section 1.6; On subspaces: pp 227-232; slides week3]

Oct 6, 2015 : Any Span is a subspace. The span is unchanged if and only if we remove one vector which is a linear combination of the others. Linear dependence. Examples.  [Finish Section 1.6 and slides Linear(in)dep ]

Oct 8, 2015: Linear independence. Basis and dimension. Dependence/independence via ranks. The maximum number of LI vectors in R^n is n. [Section 1.7 and slides Linear(in)dep ]

Oct 13, 2015: A maximal set of LI vectors is a basis. Definition of linear isomorphism.  Example of linear space (polynomials of degree  smaller or equal to n). Definition of matrix multiplication. Exercises: Review on Chapter 1, pag 87-88-89;  On scalar multiplication: numbers 1-20 on page 104.  [Slides  week5&6 , pag 361-362 and Example 3 pag 364; Matrix multiplication: pag 95,96,97]

Oct 15, 2015: Marix multiplication is not commutative: example. Diagonal matrices commute.  Linear Maps: definition and examples (linear map T_A induced by a matrix A).  Non commutativity of the product AB and composition of T_A, T_B. [Slides week5&6 and pag  97-98-99-100-101 (properties of the matrix product)]

Oct 20, 2015:  Iterations and the case study of long run population. Invertibility. Examples and properties of the inverse. Applications to linear systems with invertible coefficient matrix. [Slides week5&6 and pag 122-123-124-125; read  pages 126 and 127 ]

Exercises for Monday 26th October: all those on pag 104 and  105; 51, 52, 53, 63 and 64  on page 106; all those on page 130 and 131 (except the ones on elementary matrices)

Oct 22, 2015:  Gaussian elimination on A, with row echelon form R,  amounts to left multiplication of  an invertible matrix E with A: EA=R.  Columns  correspondence property. Application: find a basis for the Span of some given vectors. Algorithm for the computation of the inverse. [Slides week5&6 and pag 129-130 and 136]

Oct 27, 2015: Midterm.

Oct 29, 2015: Invertibility theorem. Iterates of symmetric matrices. More on linear maps, examples and  representation theorem (any linear map T is induced by a matrix A).  If T: R^n –> R^m  is linear, then  Im T is a subspace  of R^m of dimension rank A. [Slides week7 and Section 2.7 ]

Nov 3, 2015:  Ker of a linear map T. Structure theorem of the solutions of a non-homogenous system. Linear maps: surjectivity, injectivity and bijectivity seen via the rank of the associated map. [Slides week7 and Section 2.8 (excluded composition); pages 232-233-234-236-237-255 on Im T and Ker T; please note that the book calls the Ker ‘null space’ ]

Nov 5, 2015: Composition of linear map is linear, and associated matrix (the product BA) and example. Determinant of a 2×2 matrix A: definition. The determinant is non zero if and only if the matrix is invertible. Geometrically, the absolute value of the determinant is the area of the parallelogram constructed over the two columns of A. Laplace rule for the determinant of higher order matrices. The determinant of a diagonal matrix is the product of the diagonal elements. [Section 3.1 of the book and first pages of determinants]

Nov 10, 2015: Properties of the determinant. Computation of the determinant via Gaussian Elimination on a suitably obtained REF. Effect of a linear transform on an area. [Section 3.2 and 3.3 (except Cramer’s rule) and slides determinants]

Nov 12, 2015: (**optional reading not required in the exam)  Double integrals. Tonelli’s theorem. Change of variables for double integrals when the transform is linear. Bivariate probability density functions and probabilities as double integrals. [Slides integrals]

Nov 17, 2015: Eigenvectors and eigenvalues of linear endomorphisms on R^n. Eigenspaces and geometric multiplicity.  Examples. Characteristic polynomial p(x) of a square  matrix  A. The roots of p are the eigenvalues of the endomorphism induced by A. [Slides week10&11 + Chap 5]

Nov 19, 2015: Algebraic multiplicity. Eigenvalues of Upper Triangular matrices. Basis of eigenvectors when the algebraic multiplicity equals the geometric multiplicity for each eigenvalue.  [Slides week10&11]

Nov 24, 2015: Similar matrices. Spectral properties are invariant under similarity. Diagonalization. Iterates of a matrix similar to a diagonal one [Slides  week10&11]

Nov 25, 2015: Applications of eigenvalues and eigenvectors in linear, discrete dynamical systems: long run equilibria. [Final part of slides  week10&11 –  slides only, no correspondence in the book; Exercises  dyn_systems]

Nov 26, 2015: Euclidean norm and scalar product: definition and properties. Balls and spheres in R^n.  Examples  [Chap 7: Orthogonality, first 4 pages]

Dec 1, 2015: Pythagora’s theorem in R^n. Orthogonal projection on a line. Cauchy-Schwarz inequality. Triangle inequality. Orthogonal set of vectors. Orthogonal vectors are LI. Representation of a vector in terms of an orthogonal basis.

Dec 3, 2015: The Gram-Schmidt algorithm.  [Chapter 7 up to page 443]. Revision.