Introduction

Linear algebra has spectacular applications in disciplines as apparently diverse as computing, engineering and biology; moreover, it's at the core of much of pure mathematics. This course provides an introduction to linear algebra. As befits a 300-level course, it will involve a mixture of computation and proof, and of theory and application.

Logistics

Professor Jeff Achter
j.achter@colostate.edu
Weber 216, 491-6716
Lecture: MWF 1-1:50PM, Engrg E104
Office hours: TBA
Textbook: There is a recommended textbook, and a free, on-line text:
- Recommended: Elementary Linear Algebra (Applications Version), Howard Anton and Chris Rorres, Wiley, 2010.
- Freely available: A First Course in Linear Algebra, by Robert Beezer.
More information about textbooks.

Requirements and other expectations

Homework and classwork -- 30% Homework will be assigned weekly, and is due at the beginning of class on Fridays. No late homework will ever be accepted. Your work must be neat, and the pages stapled, in order to be graded.
Midterms -- 40% There will be in-class exams on Friday, March 7 and Monday, April 14.
Final -- 30% The final exam for this course will be held on Thursday, May 15, 11:50AM-1:50PM.
No makeup exams will be given; you must take each exam as scheduled. You must pass the final in order to pass the course.

Help

This is challenging material, and you may need a little help every now and then.

Questions directed to j.achter@colostate.edu will be answered swiftly.
Keep an eye on the news section of the page for more resources.

Topics

We will address systems of linear equations; matrix algebra; vector spaces and bases; linear transformations; eigenvalues and eigenvectors; and inner products. Our approach will mix theory and computation, and in particular will sometimes be more abstract than that of the officially recommended textbook.

As the course unfolds, the following table will indicate the relevant sections of the two textbooks.

Week Topic Beezer Anton and Rorres

1
1/20 Systems of linear equations
examples and applications; row reduction; echelon forms;
associated(augmented) matrices WILA What is Linear Algebra?
SSLE Solving Systems of Linear Equations
RREF Reduced Row-Echelon Form 1.1, 1.2

2
1/27 Gaussian elimination
More on echelon forms; Gaussian elimination; solution sets of linear equations. TTS Types of solution sets 1.2

3
2/3 Matrices and vectors
(column) vectors; matrices; sums; legal products VO Vector Operations
MO Matrix Operations
MM Matrix Multiplication
1.3

4
2/10 Inverses, span
Definition of inverse; computing inverses; elementary row operations and elementary matrices; spans and linear combinations. MISLE Matrix Inverses
LC Linear Combinations
SS Spanning Sets 1.4, 1.5, 1.6, 4.2(*)

5
2/17 Vector spaces Abstract vector spaces, subspaces. VS Vector Spaces
S Subspaces
4.1, 4.2

6
2/24 Linear combinations Linear dependence/independence, spanning sets, basis, dimension. LI Linear Independence
LISS Linear Independence and Spanning Sets
B Bases 4.3, 4.4

7
3/3 Midterm

8
3/10 Linear transformations LT Linear Transformations 8.1

9
3/24 Linear transformations Null space and image; rank-nullity theorem; bases and matrices MR Matrix Representations
CB Change of Basis
ILT and SLT 8.4, 8.3, 8.5

10
3/31 Bases and matrices continued

11
4/7 Minimal polynomial definition, existence, roots Notes

12
4/14 Midterm
Determinants determinant as volume; properties. DM Determinant of a Matrix
PDM Properties of Determinants of Matrices

13
4/21 Eigenvectors eigenvalues, bases of eigenvectors, diagonalizability. EE Eigenvalues and Eigenvectors
SD Similarity and Diagonalization

14
4/28 Characteristic and minimal polynomials Inner products

15
5/5 Geometry of inner products norms, estimates, orthonormal bases O Orthogonality

(*) means the subject in question has better coverage in the online textbook.

This page is available at http://www.math.colostate.edu/~achter/369

Week	Topic	Beezer	Anton and Rorres
1 1/20	Systems of linear equations examples and applications; row reduction; echelon forms; associated(augmented) matrices	WILA What is Linear Algebra? SSLE Solving Systems of Linear Equations RREF Reduced Row-Echelon Form	1.1, 1.2
2 1/27	Gaussian elimination More on echelon forms; Gaussian elimination; solution sets of linear equations.	TTS Types of solution sets	1.2
3 2/3	Matrices and vectors (column) vectors; matrices; sums; legal products	VO Vector Operations MO Matrix Operations MM Matrix Multiplication	1.3
4 2/10	Inverses, span Definition of inverse; computing inverses; elementary row operations and elementary matrices; spans and linear combinations.	MISLE Matrix Inverses LC Linear Combinations SS Spanning Sets	1.4, 1.5, 1.6, 4.2(*)
5 2/17	Vector spaces Abstract vector spaces, subspaces.	VS Vector Spaces S Subspaces	4.1, 4.2
6 2/24	Linear combinations Linear dependence/independence, spanning sets, basis, dimension.	LI Linear Independence LISS Linear Independence and Spanning Sets B Bases	4.3, 4.4
7 3/3	Midterm
8 3/10	Linear transformations	LT Linear Transformations	8.1
9 3/24	Linear transformations Null space and image; rank-nullity theorem; bases and matrices	MR Matrix Representations CB Change of Basis ILT and SLT	8.4, 8.3, 8.5
10 3/31	Bases and matrices continued
11 4/7	Minimal polynomial definition, existence, roots	Notes
12 4/14	Midterm Determinants determinant as volume; properties.	DM Determinant of a Matrix PDM Properties of Determinants of Matrices
13 4/21	Eigenvectors eigenvalues, bases of eigenvectors, diagonalizability.	EE Eigenvalues and Eigenvectors SD Similarity and Diagonalization
14 4/28	Characteristic and minimal polynomials Inner products
15 5/5	Geometry of inner products norms, estimates, orthonormal bases	O Orthogonality