Math 4242 :

**Instructor**: Willard
Miller

Office: Vincent Hall 513

Office Hours: 11:15-12:05 MW, 9:05-9:55 F, or by appointment

Phone: 612-624-7379

miller@ima.umn.edu,
miller@math.umn.edu

**Prerequisites**: Previous exposure to linear algebra: determinants
and Cramer's rule.

**Textbook**: G. Strang , Linear Algebra and Its Applications, 3rd
Ed., Brooks/Cole (Thomson Learning), 1988.

**Class Description:** A foundation course in linear algebra, with
applications. Topics include: linear transformations, vector spaces, matrix
calculus, solutions of systems of linear equations, determinants, orthogonality,
LDU decompositions, SVD decompositions, canonical forms. Applications include:
Gram-Schmidt process, least-squares approximations, the Fast Fourier transform
(FFT), etc. This material is basic, both for the understanding of the theory
of linear algebra and for numerical computation.

**Software**: MATLAB (by The Mathworks, Inc.) is ideal, but not required.
No previous exposure is expected.

**Policies**:

- Eight homework assignments [NO late homework will be accepted without good excuses].
- Two Midterms plus the Final Exam [NO laptops. Standard calculator is OK. NO makeup tests without rigorous emergency reasons. Athletes please present "Proofs of Activities" in advance].
- Grading Policy: Homework (20%), Two Midterms (40%), Final Exam (40%)

**Content and Style**: Will try to cover (inclusively) up to Chapter
6 (out of 7 chapters). Approximately two to three weeks for each chapter.
Homework assignments are mainly from the textbook, which has many well
designed exercise problems. The theory will predominate, but there will
be considerable attention to applications in other fields.

**Student Conduct:** Statement on Scholastic Conduct: Each
student should read the college bulletin for the definitions and possible
penalties for scholastic dishonesty. Students
suspected of cheating will be reported to the Scholastic Conduct Committee.

**Paper Grader:** Hazem Hamdan

Office: Vincent Hall 522

Office Hours: 12:20-1:20 MW,

Phone: 612-624-4143

hamdan@math.umn.edu,

**Introduction to MATLAB (courtesy
of Professor Peter Olver) **Postscript
file PDF
file

Week |
Topic |
Section |
HW due |
||

W | JAN 22 | 1 | Introduction and Linear equations | 1.1-1.2 | |

F | JAN 24 | 1 | Introduction to Gaussian Elimination | 1.3 | |

M | JAN 27 | 2 | Matrix Notation and Multiplication | 1.4 | |

W | JAN 29 | 2 | Triangular Factorization | 1.5 | |

F | JAN 31 | 2 | Inverses and Transposes | 1.6 | |

M | FEB 3 | 3 | Inverses and Transposes/Applications | 1.6/1.7 | 1 |

W | FEB 5 | 3 | Applications | 1.7 | |

F | FEB 7 | 3 | Vector Spaces and Subspaces | 2.1 | |

M | FEB 10 | 4 | m Equations in n Unknowns | 2.2 | |

W | FEB 12 | 4 | Linear Independence, Basis, & Dimension | 2.3 | 2 |

F | FEB 14 | 4 | Row, Column, and Null Spaces | 2.4 | |

M | FEB 17 | 5 | Row, Column, and Null Spaces | 2.4 | |

W | FEB 19 | 5 | Linear Transformations | 2.6 | |

F | FEB 21 | 5 | Orthogonal Subspaces | 3.1 | |

M | FEB 24 | 6 | Orthogonal Subspaces | 3.1 | 3 |

W | FEB 26 | 6 | Inner products, Projections | 3.2 | |

F | FEB 28 | 6 | Inner products, Projections | 3.2 | |

M | MAR 3 | 7 | Least Squares Approximations | 3.3 | |

W | MAR 5 | 7 | Orthogonal Bases, Gram-Schmidt Orthogonalization | 3.4 | 4 |

F | MAR 7 | 7 | Orthogonal Bases, Gram-Schmidt Orthogonalization | 3.4 | |

M | MAR 10 | 8 | Fast Fourier Transform | 3.5 | |

W | MAR 12 | 8 | Review | 3.6 | |

F | MAR 14 | 8 | Midterm I: Chapters 1-3 |
||

Spring Break |
|||||

M | MAR 24 | 9 | Properties of the Determinant | 4.1-4.2 | |

W | MAR 26 | 9 | Formulas for the Determinant | 4.3 | |

F | MAR 28 | 9 | Applications of Determinants | 4.4 | |

M | MAR 31 | 10 | Introduction to Eigenvalues | 5.1 | 5 |

W | APR 2 | 10 | Matrix Diagonalization | 5.2 | |

F | APR 4 | 10 | Powers of Matrices | 5.3 | |

M | APR 7 | 11 | Powers of Matrices | 5.3 | |

W | APR 9 | 11 | Matrix Exponential | 5.4 | |

F | APR 11 | 11 | Matrix Exponential | 5.4 | |

M | APR 14 | 12 | Complex Matrices | 5.5 | 6 |

W | APR 16 | 12 | Review | ||

F | APR 18 | 12 | Midterm II: Chapter 4, Sections 5.1-5.4 |
||

M | APR 21 | 13 | Complex Matrices | 5.5 | |

W | APR 23 | 13 | Similarity Transformations | 5.6 | |

F | APR 25 | 13 | Similarity Transformations | 5.6 | |

M | APR 28 | 14 | Extreme Values and Positive Definite Matrices | 6.1 | 7 |

W | APR 30 | 14 | Tests for Positive Definiteness | 6.2 | |

F | MAY 2 | 14 | Tests for Positive Definiteness | 6.2 | |

M | MAY 5 | 15 | Indefinite Matrices | 6.3 | |

W | MAY 7 | 15 | The Singular Value Decomposition (SVD) | Appendix A | 8 |

F | MAY 9 | 15 | Review | ||

TU | MAY 13 | Final Exam, 8:00 -10:00 am, VH2 |

**Homework Assignments**

**HW1 **: Due in class: Monday, February 3

1.2.3, 1.2.5 (page 10); 1.3.1, 1.3.5(page16); 1.4.1, 1.4.5, 1.4.22
(pages 27-30), 1.5.4, 1.5.5 (page 39)

**HW2**: Due in class: Wednesday, February 12

1.5.9, 1.5.11(page 40); 1.6.3, 1.6.8, 1.6.19 (pages 49-51); 2.1.3,
2.1.5 (page 69), 2.2.3, 2.2.7 (pages 77-78);

**HW3 **Due in class: Monday, February 24

2.3.1, 2.3.5, 2.3.16 (pages 87-88); 2.4.3(page 99), 2.6.7, 2.6.8, 2.6.18(pages
125-126)

**HW4** Due in class: Wednesday, March 5

3.1.1,3.1.5 (pages 141-142); 3.2.3, 3.2.8(page 151); 3.3.3,
3.3.5, 3.3.7, 3.3.12, 3.3.26 (pages 163-165).

**HW5** Due in class: Monday, March 31

3.4.2,3.4.6, 3.4.13, 3.4.17(pages 180-181),4.2.1,4.2.4,4.2.6(page 219),
4.3.5, 4.3.6, 4.3.7(pages 228-229),4.4.5(page 238)

**HW6 **Due in class: Monday, April 14

5.1.1,5.1.2,5.1.3(pages 251-252), 5.2.2, 5.2.7, 5.2.10(pages
260-261),5.3.1, 5.3.9(pages 272-273),5.4.1, 5.4.2(page 286)

**Solutions to HW6 ** PDF

**HW7** Due in class: Monday, April 28

5.5.1, 5.5.7, 5.5.8(pages 301-302),5.6.3,5.6.5 5.6.13(pages 315-316)

**HW8** Due in class: Wednesday, May 7

6.1.2, 6.1.4(page 328), 6.2.1, 6.2.2,6.2.4,6.2.7,6.2.12(page 337),6.3.2,
6.3.11(pages 345-346)