| TIMETABLE |
LECTURES |
EXERCISES |
EXAMS |
| Instructor: |
Prof. Giuseppe Pareschi |
| Teaching Assistant: |
Dr. Pietro Sabatino |
| Textbook: |
T. Apostol, Calculus, 2nd edition, Wiley. Vol.I (Chapters from 12 to 16), Vol. II (Chapters from 3 to 5) |
| MON |
9.15 - 11.15: Tutoring Linear Algebra and Geometry (Pareschi). 11.30 - 13.15 Linear Algebra and Geometry (Pareschi) |
| WED |
14.00 - 15.45 Mathematica Analysis II (Bertsch) |
| FRI |
14.00 - 15.45 Mathematical Analisys II (Bertsch) 16 - 18 Tutoring (Bertsch) |
| Oct. 15 |
The space V(n). Vector addiction. Multiplication by scalars |
|
| Oct. 18 |
Geometric interpretation of Vector addition and multiplication by scalars. |
Assigned Exercises: 12.4 p. 450-451 Vol. I |
| Oct. 22 |
Dot product. Norm. Orthogonality. |
Assigned Exercises: 12.8 p. 456-457 Vol. I |
| Oct. 25 |
Orthogonal projection. Angle between two vectors. |
Assigned Exercises: 12.11 p. 460-462 Vol. I |
| Oct .29 |
Linear combinations. The linear span of a set of vectors. Linear independence. |
|
| Nov. 5 |
Linear independence (continuation). Bases of V(n). |
Assigned Exercises: 12.15 p. 467-468 Vol. I |
| Nov. 8 |
Lines in V(n). Parametric equations. |
|
| Nov. 12 |
Distance point line in V(n). Normal vector to a line in V(2). Cartesian equation of a line in V(2). Distance point line in V(2). |
Assigned Exercises: 13.5 p. 477 Vol. I |
| Nov. 15 |
Planes: parametic equations
|
Assigned exercises: 13.8 p. 482-483 Vol. I |
| Nov. 19 |
Determinants of order two and three. Cross product in V(3) | |
| Nov. 22 |
Scalar triple product in V(3). Cramer's rule for linear systems of three equations and three
unknowns. |
Assigned exercises: 13.14 p. 491-493 |
| Nov. 26 |
Linear systems of three equations in three unknowns (continuation). Planes in V(3): normal
vectors, cartesian equations, distance point-plane |
Assigned exercises: 13.17 p. 496-497 |
| Nov. 29 |
Conics: definition, basic equation and polar equation. |
Assigned exercises: 13.21 p. 503 |
| Dec. 3 |
Conics symmetric about the origin: basic equation |
|
| Dec. 6 |
Ellipses and Hyperbola with center at the origin. Cartesian equation in
standard form. |
|
| Dec.13 |
Ellipses, hyperbolas, parabolas. |
Assigned exercises: 13.24, 13.25 p.508-512 |
| Dec. 17 |
Vector valued funcions: algebraic operations, derivation rules. Applications to
curves. Equivalent vector-valued functions. Tangent vector and tangent line. |
Assigned exercises: 14.4 p.516-517 and examples at p.519-520
Supplementary exercises on the material of Chapters 12 and 13: pdf |
| Dec. 20 |
Applications to curvilinear motion. Velocity, speed and acceleration.
Unit tangent vector and unit normal vector. Osculating plane. |
Assigned exercises: 14.7 p. 524-525 and 14.9 p. 528-529 |
| Jan. 10 |
Arc-length. Anc-length function. Arc-length reparametrization.
Curvature. |
Assigned exercises: 14.13 p. 535-536 |
| Jan. 14 |
Characterization of lines and circles. |
Assigned exercises: 14.15 p.538-539 |
| Jan 17 |
Formulas for velocity and acceleration in polar coordinates. Motion
with radial acceleration: the position vector sweeps area at constant rate. Proof of Kepler's
laws. |
Assigned exercises: 14.19 p. 543-545 and 14.21 p. 549-550 |
| Jan 21 |
Exercises. |
|
| Jan 24 |
Exercises. |
|
| Jan 28 |
Exercises |
|
| Feb 28 |
Real linear spaces. Examples, including function spaces. Complex linear spaces. Linear combinations.
Linear span. |
Assigned exercises: 15.5 p.555-556 |
| March 2 |
Linear dependence and independence. Spanning subsets. Finite/infinite-dimensional linear spaces.
Bases |
|
| March 7 |
Bases (continuation). Inner products for real linear spaces. Examples (including inner products on
function spaces) |
Assigned exercises: 15.9 p. 560-561 |
| March 9 |
Inner product for complex linear spaces. Cauchy-Schwartz inequality. Norms. Orthogonality.
Orthogonality and Independence. Orthonormal bases. Components with respect to orthonormal bases. Inner product
with respect to orthonormal bases. Gram-Schmidt orthogonalization. | Assigned exercises:
15.12 p. 566-568 |
| March 9 |
Inner product for complex linear spaces. Cauchy-Schwartz inequality. Norms. Orthogonality.
Orthogonality and Independence. Orthonormal bases. Components with respect to orthonormal bases. Inner product
with respect to orthonormal bases. | Assigned exercises:
15.12 p. 566-568 |
| March 14 |
Orthogonality of the set of trigonometric polynomials. Gram-Schmidt orthogonalization. Legendre polynomials. Orthogonal
complement of a linear subspace. Orthogonal
decomposition theorem for finite-dimensional linear subspaces.
|
|
| March 16 |
Orthogonal projection onto a finite-dimensional linear subspace. Distance from a finite-dimensional linear subspace. Nearest
element. Best approximation of functions with function in a given finite-dimensional linear subspace (examples: the space of trogonometric
polynomials of order at most n, the space of polynomials of oprder at most n). Propoerties of the orthogonal complement in a finite-dimensional
euclidean linear space:Supplementary notes and exercises | Assigned exercises: 15.17 p.
576-577 |
| March 21 |
Linear trasformations. Examples. Null-space and range. Nullity + rank theorem.
| Assigned exercises: 16.4 p.582-583
|
| March 23 |
Linear transformations: geometric examples (rotation about the origin and reflection with
respect to a line trough the origin in the plane). Injectivity, surjectivity, bijectivity,
invertibility for linear transformations.
| Assigned exercises 16.8 p. 589-590
|
| April 4 |
Examples of null-space and range in finite and infinite dimension. Linear transformation with prescribed values of the vectors of a given basis.
|
|
| April 11 |
Linear transformations with prescribed values of the vector of a given basis (cont.). Matrices. Matrix multiplication.
| Assigned exercises: 16.16 p.604-605
|
| April 13 |
The linear transformation from V(n) and V(m) associated to a matrix with m rows and n columns. Rank of a matrix: the maximal mumber of independent columns
equals the maximal number of independent rows.
| Assigned exercises: 16.12 p. 596-597.
|
| April 17 |
Computing the rank of a matrix and solving systems of linear equations via row elimination (Gauss' algorithm). Rouche'-Capelli theorem.
Correspondence between matrix multiplication and composition of linear transformations. Invertible matrices and their inverses. Characterization of invertible
matrices as the non-singular ones (namely those of maximal rank). | Assigned exercises: 16.20 p. 613-614.
|
| April 20 |
Computing the inverse matrix via row elimination. Solving matrix equationx of the forma AX=B (where A and B are suitable matrices). Inverse matrices
and linear systems. Inverse matrices and change of coordinates.
Supplementary notes and exercises
| Assigned exercises: 16.21 p. 614-615.
|
| May 2 |
Determinant functions as multilinear and alternating functions. Diagonal and upper triangular matrices. Computation of multilinear and
alternating functions
via row elimination.
| Assigned exercises: 3.6 p. 79-81 Vol. II
|
| May 16 |
Cramer's rule. Eigenvalues and eigenvectors. Eigenspaces. Characteristic polynomial.
| Assigned exercises: 4.4 p.101 Vol. II
|
| May 18 |
Generalities on real and complex polynomials. Trace and determinant as coefficients of the characteristic polynomials.
Independence of eigenvectors belonging to distinct eigenvalues.
| Assigned exercises: 4.8 p.107-108 Vol. II
|
| May 23 |
Examples of matrices with double or more generally multiple eigenvalues. Matrix reprensenting the same transformation with respect to different bases: similar matrices.
The characteristic polynomial of matrices representating a given linear transformation with respect to different bases is the same. The dimension of the eigenspace does not excedd
the multiplicity of the eigenvalue. Diagonalizability and diagonalization of linear transformations and matrices | Assigned exercises: 4.10 p. 112-113 Vol. II
|
| May 25 |
Hermitian and skew-hermitian transformations. The real case: symmetric and skew-symmetric transformations. hermitian, skew-hermitian, symmetric, and skew-symmetric
matrices. Matrix representation: a transformation is (skew-)hermitian if and only if its representative matrix with respect to an orthonormal basis is (skew-)hermitian. The
eigenvalues of a hermitian (or symmetric) transformation are all real. The eigenvalues of a skew-hermitian (or skew-symmetric) transformation are either zero or pure-imaginary.
Eigenvectors velonging to distinct eigenvalures are always orthogonal.
| Assigned exercises: 5.5 p. 118-120 Vol. II
|
| May 27 |
Examples of hermitian, symmetric, skew-hermitian and skew-symmetric transformations and their eigenvalues and eigenvectors (both in finite and infinite dimension)
|
|
| May 30 |
Spectral theorem: a hermitian or skew hermitian transformation on afinite-dimensional linear space has always an orthonormal basis of eigenvectors. Unitary and
orthogonal basis. Unitary and orthogonal base-change. Real quadratic forms. | Assigned exercises: 5.11 p. 124-126 Vol. II
|
| June 6 |
Reduction of a real quadratic form to diagonal form by means of an orthonormal basis of eigenvector of the corresponding symmetric matrix. Application to conics. Extremal values of a real quadratic
forms on the unit sphere.
|
Assigned exercises: 5.15 p. 134-135 Vol.II. Supplementary exercises
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| June 13 |
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