Prof. Carlos F. Borges
349 Glasgow Hall
656-2124
borges@nps.navy.mil
Students should have mathematical background at the level generally expected of someone with a B.S. in Engineering, i.e. familiarity with Calculus and solid algebra skills.
The fundamental algebra of vectors and matrices including addition, scaling, and multiplication. Block operations with vectors and matrices. Algorithms for computing the LU (Gauss) factorization of an nxm matrix, with pivoting. Matrix representation of systems of linear equations and their solution via the LU factorization. Basic properties of determinants. Matrix inverses. Linear transformations and change of basis. The four fundamental subspaces and the fundamental theorem of linear algebra. Eigenvalues and eigenvectors.
The grade will be based on two midterms, a final, and several computer projects. Homework will be assigned but will not be graded.
|
Hours |
Topic |
Sections |
|
|
Introduction
to Vectors |
1.1 - 1.2 |
|
|
Solving
Linear Equations |
2.1 - 2.5 |
|
|
LU
Factorization |
2.6 - 2.7 |
|
|
Vector
Spaces |
3.1 |
|
|
The Null
Space of A |
3.2 |
|
|
Rank and
the Reduced Echelon Form |
3.3 |
|
|
The
Complete Solution to Ax = b |
3.4 |
|
|
Independence,
Basis, and Dimension |
3.5 |
|
|
The Four
Fundamental Subspaces |
3.6, 4.1 |
|
|
Projections |
4.2 |
|
|
Linear
Least Squares Problems |
4.3 |
|
|
Orthogonal
Bases and Gram-Schmidt |
4.4 |
|
|
Determinants |
5.1 - 5.3 |
|
|
Eigenvalues
and Eigenvectors |
6.1 |
|
|
Diagonalization |
6.2 |
|
|
Symmetric
and Positive Definite Matrices |
6.4 – 6.5 |
|
|
Linear
Transformations |
7.1 |
|
|
The Matrix
of a Linear Transformation |
7.2 |
|
|
Change of
Basis |
7.3 |
The following is a list of suggested homework problems. Skill, understanding, and proficiency are gained only by practice so you should consider the problems on this list an absolute minimum requirement.
|
Section |
Problems |
|
1.1 |
1-5, 8, 9 |
|
1.2 |
1-6, 9 |
|
2.1 |
10-21 |
|
2.2 |
1-15 |
|
2.4 |
1-6, 11, 13, 14, 18, 19, 21, 26-29 |
|
2.5 |
1, 2, 4-6, 8a, 10-12, 15, 25, 27 |
|
2.6 |
13-16 |
|
2.7 |
1, 2, 4, 7, 11, 16, 22, 24 |
|
3.1 |
1, 2, 4, 5, 9, 10, 19, 20 |
|
3.2 |
1-3, 5, 9, 21-24 |
|
3.4 |
1, 2, 4, 16-19, 23 |
|
3.5 |
1-5, 7, 9, 12, 13, 17 |
|
3.6 |
1-4, 7, 9, 10 |
|
4.1 |
3-7,10 |
|
4.2 |
1, 3, 5-7, 11-13 |
|
4.3 |
1, 3-7 |
|
4.4 |
1-3, 5-7, 10b, 11, 13-18 |
|
5.1 |
13-19 |
|
6.1 |
2-6, 9, 11, 13 |
|
6.2 |
1-4, 8 |
|
6.4 |
3-5 |
|
6.5 |
2, 6, 7 |
|
7.1 |
1, 3, 5, 6, 8, 10, 12 |
|
7.2 |
1-7, 11, 15 |