Time: 9:40-10:30 NS 205

Instructor: Prof. Nick Korevaar

Web Page:http://www.math. utah.edu/~korevaar

Office: JWB 218

Telephone: 581-7318

Email:korevaar@math.utah.edu

Office hours: M 2-2:50 p.m., T 12:30-2:30 p.m., W 2-2:50 p.m., F 8:20-9:00 a.m.

The course begins by studying linear systems of equations and the Gauss-Jordan method of systematically solving them. We see how to write these problems more succinctly in matrix form, the algebra of matrix operations, about inverses of non-singular square matrices, about determinants and their usefulness in solving linear problems. These topics comprise chapters 1-2 of the text.

In chapter 3 we review the linear geometry of
**R**^{2},**R**^{3}, and
**R**^{n
} and discuss the geometric meaning of matrices and
determinants, as well as the
dot and cross products and their geometric meanings.
After chapter 3 we skip to the application called linear programming
(chapter 7), which is used heavily in business.

At this point the course takes a turn towards the abstract as we
study general vector spaces in chapters 4 and 6.
Basically a vector space is a collection of objects (called vectors)
which you can add and scalar multiply, such that certain arithmetric
properties hold. From these arithmetric properties one develops
notions such as linear independence, bases, dimension, subspaces,
coordinates with respect to a basis, change of basis, linear
transformations between vector spaces, kernel and range subspaces of
linear transformations. We usually visualize vectors in the concrete
example of **R**^{n}, but in fact there are very natural spaces of
functions and
of solutions to certain (homogeneous linear)
differential equations which are also vector spaces, so that these
abstract concepts also apply to them. It is precisely because vector
spaces appear in these different disguises that it is worthwhile to
discuss them in this abstract way: as characterized by properties
rather than by explicit descriptions. You will appreciate this more
when you take Math 2280 and apply vector space theory to your study of
linear differential
equations.

We will discuss the notion of eigenvectors and eigenvalues for
linear transformations from **R**^{n} to **
R**^{n},in chapter. These will also be used heavily in
Math 2280.

The dot product in **R**^{n}
lets one talk about orthogonality and orthogonal projections, and we
discuss several applications related to this circle of ideas:
Gram-Schmidt orthogonalization, methods of least squares (8.4),
diagonalizing quadratic forms (8.8), rotating space to express conic
sections and quadratic surfaces optimally (8.9-8.10).

We will show there is also a natural ``dot'' product on certain function spaces, and use this to motivate Fourier series (appendix B.1) which you will return to in Math 2280.

Although we will use mostly real (scalar) vector spaces, it is often important in applications to allow complex numbers as your scalars, and we discuss this in Appendix A. If there is time at the end of the semester we will also spend a couple of lectures sketching some of the applications of our work to differential equations (8.6), as a way of forshadowing your work in Math 2280.

There will be approximately 6 computer projects during the semester, to enhance and expand upon the material in the text. They will be written in the software package MAPLE. On MAPLE days we will meet in the Math Department Computer Lab located in Building 129, between JWB and LCB. We do not assume you have had any previous experience with this software and we will make the necessary introductions during the first visit to the lab.

## Tentative Daily Schedule | |||

## exam dates fixed, daily subject matter approximated | |||

F | 28 Aug | 1.1 | linear systems |

M | 31 Aug | 1.2 | matrices |

T | 1 Sept | 1.3 | matrix multiplication |

W | 2 Sept | 1.4 | matrix operations |

F | 4 Sept | 1.5 | solving linear systems |

M | 7 Sept | none | Labor Day |

T | 8 Sept | maple | project I |

W | 9 Sept | 1.6 | matrix inverses |

F | 11 Sept | 2.1 | determinant definition |

M | 14 Sept | 2.2 | cofactor expansions and applications |

T | 15 Sept | 2.2 | " " |

W | 16 Sept | 3.1 | vectors in the plane |

F | 18 Sept | 3.2 | n-vectors |

M | 21 Sept | 3.3-3.4 | linear transformations |

T | 22 Sept | maple | project II |

W | 23 Sept | 3.3-3.4 | linear transformations |

F | 25 Sept | 3.5 | cross product |

M | 28 Sept | 3.6 | lines and planes |

T | 29 Sept | 7.1 | linear programming problem |

W | 30 Sept | 7.2 | simplex method |

F | 2 Oct | 7.2 | " " |

M | 5 Oct | review | review |

T | 6 Oct | exam 1 | 1-3, 7.1-7.2 |

W | 7 Oct | 4.1 | real vector spaces |

F | 9 Oct | none | fall break day |

M | 12 Oct | 4.2 | subspaces |

T | 13 Oct | 4.3 | Linear Independence |

W | 14 Oct | 4.4 | basis and dimension |

F | 16 Oct | 4.4 | basis and dimension |

M | 19 Oct | 4.5 | homogeneous systems |

T | 20 Oct | maple | project III |

W | 21 Oct | 4.6 | matrix rank |

F | 23 Oct | 4.7 | coordinates, change of basis |

M | 26 Oct | 4.8 | orthonormal bases |

T | 27 Oct | 4.8-4.9 | orthogonal complements |

W | 28 Oct | 4.9 | orthogonal complements |

F | 30 Oct | 8.4 | least squares |

M | 2 Nov | 8.4 & B.1 | inner product spaces IV |

T | 3 Nov | maple | project IV |

W | 4 Nov | B.1 | Fourier series |

F | 6 Nov | A.1 | Complex numbers |

M | 9 Nov | A.2 | Complex linear algebra |

T | 10 Nov | 6.1 | linear transformations |

W | 11 Nov | 6.2 | kernel and range |

F | 13 Nov | 6.3 | matrix of linear transformation |

M | 16 Nov | 6.3 | continued |

T | 17 Nov | review | review |

W | 18 Nov | exam 2 | 4, 8.4,A,B.1,6 |

F | 20 Nov | 5.1 | eigenvalues and eigenvectors |

M | 23 Nov | 5.2 | diagonalization continued |

T | 24 Nov | maple | project V |

W | 25 Nov | 5.2 | diagonalization continued |

F | 27 Nov | none | Thanksgiving |

M | 30 Nov | 8.8 | quadratic forms |

T | 1 Dec | maple | project VI |

W | 2 Dec | 8.9-8.10 | conic sections |

F | 4 Dec | 8.9-8.10 | quadric surfaces |

M | 7 Dec | 8.10 | quadric surfaces |

T | 8 Dec | 8.6 | differential equations |

W | 9 Dec | 8.6 | differential equations |

F | 11 Dec | all | entire course |

T | 15 Dec | FINAL EXAM | entire course 10:00-12:00 |