As part of its assessment activities in the 1998-99 academic year, the UCC
asked Professor David Stanford to organize a faculty discussion of the
appropriate content of Math 211. All instructors in Math 211 between Fall
1995 and February 1999 were invited to join the discussion, and Professors
Drew, Rodman, Mathias, Johnson, Stanford, Rublein, Spitkovsky, Yopp,
Bolotnikov, and Fallat participated in the discussions and e-mail 
exchanges.

In his e-mail to UCC dated February 19, 1999, Professor Stanford reported
the consensus of the discussion group to be as follows. As organized in
the spring of 1999, our Math 211 course covered virtually all of the "Core
Syllabus" in the Linear Algebra Curriculum Study Group (LACSG) report
(quoted at the end of this section), as well as some things listed under
"Supplementary Topics," e.g. an introduction to abstract vector spaces and
linear transformations thereon, and sometimes positive definite matrices.

In a report to UCC dated April 21, 1999, Professor Stanford answered a
series of questions posed to the discussion group by UCC. The questions
and answers were as follows:

1) Does the Math 211 discussion group agree that the core syllabus of
LACSG should be included in every 211 course? YES, THE GROUP AGREES.

2) Does the Math 211 discussion group agree that a Math 211 course 
consisting of the entire LACSG core syllabus and nothing else would be an
acceptable course? NO, THE GROUP DISAGREES.

3) Does the Math 211 discussion group agree that the core LACSG syllabus
can be covered in about 70% of the time available in a 3-credit Math 211?
EIGHT OF THE NINE VOTING PARTICIPANTS AGREE.

4) Does the Math 211 discussion group agree that the current Math 211
course (as usually taught by most discussion particpants) covers the LACSG
core syllabus plus a reasonable selection of supplementary topics? If so,
does that mean that our current Math 211 syllabus spends about 80% of the
time on the LACSG core topics and the rest spent on additional topics such
as those suggested by LACSG? YES, THE DISCUSSION GROUP AGREES.
 

In May of 2003, the group of recent Math 211 instructors reaffirmed the 
decisions of the 1999 discussion group and added the recommendation that 
examples and problems using complex scalars be included in Math 211.

**************** 

The LACSG Syllabus for the First Linear Algebra Course
 

The following is essentially quoted from the report of the Linear Algebra
Curriculum Study Group that appears in "Resources for Teaching Linear
Algebra," edited by David Carlson, Charles Johnson, David Lay, Duane
Porter, Ann Watkins, and William Watkins, MAA Notes Volume 42,
Mathematical Association of America, 1997, pages 53-58. In what follows 
"day" means a "fifty-minute class period."
 

LACSG Core Syllabus

I. Matrix Addition and Multiplication (3 days)

This includes the normal topics of matrix addition, scalar multiplication,
matrix multiplication, transposition, and their algebraic properties such
as associativity of matrix multiplication.  Operations with partitioned
matrices. Motivate matrix multiplication and carefully examine three views
of the product AB:

1) Ax is a linear combination of the columns of A, with coefficients from
x; each column of AB is obtained by multiplying A by the corresponding
column of B.  Thus each column of AB is a linear combination of the
columns of A, with coefficients from the corresponding column of B.  If D
is a diagonal matrix, then AD is a scaling of the columns of A.  If P is a
permutation matrix, than AP is a permutation of the columns of A.

2) Similarly, the rows of AB are linear combinations of the rows of B.

3) AB is a sum of outer products (i.e., rank 1 matrices): AB =
col(1,A)row(1,B) + ... + col(k,A)row(k,B), when A is m by k and B is k by
n.
 

II. Systems of Linear Equations (4 days)

Gaussian elimination/elementary matrices.  Echelon and reduced echelon
form. Existence/uniqueness of solutions. Matrix inverses. Row reduction
interpreted as an LU factorization.
 

III. Determinants (2-3 days)

Determinants are readily encountered when solving 2 by 2 and 3 by 3
general linear systems.  The elementary properties of determinants are
easily discovered or illustrated using the resulting expressions.  Formal
verifications in most cases should be avoided.  Explore the use of
determinants as well as the difficulties in computing them.  Main topics:
cofactor expansion, determinants and row operations, detAB = detAdetB, and
Cramer's Rule (to show the sensitivity of solutions of Ax = b).

IV. Properties of R^n (7-8 days)

Introduce R^n as a set of n-tuples and not as a formal vector space.  
Define vector addition and scalar multiplication, but it is not necessary
to prove formally all of the properties of vector addition and scalar
multiplication.  there should be a strong geometric emphasis in the
presentation of this material.

1. Linear combinations: linear dependence and independence.

2. Bases of R^n

3. Subspaces of R^n: spanning set, basis dimension, row space and column
space (range of A as a mapping)

4. Matrices as linear transformations

5. Rank: row rank = column rank, products, connections with invertible
sub-matrices.

6. Systems of equations revisited: solution theory, rank+nullity = number
of columns.

7. Inner product: length and orthogonality, orthogonal/orthonormal sets
and bases, orthogonal matrices.
 

V. Eigenvalues and Eigenvectors (6 days)

Eigenvalues are important in a wide variety of applications. Sufficient
time should be allowed for complete coverage of this topic.  Eigenvectors
may be introduced and/or motivated using geometric examples.

1. The equation Ax = kx.

2. The characteristic polynomial and identification of some of its
coefficients (trace, determinant), algebraic multiplicity of eigenvalues.

3. Eigenspaces, geometric multiplicity.

4. Similarity: distinct eigenvalues and diagonalization (with emphasis on
AP = PD).

5. Symmetric matrices: orthogonal diagonalization, quadratic forms.
 

VI. More on Orthogonality (4 days)

Include the standard topics with strong geometric emphasis: orthogonal
projection onto a subspace; Gram-Schmidt orthogonalization and
interpretation as a QR factorization; and the least square solutions of
inconsistent linear systems, with applications to data-fitting.
 

***The total time spent on core topics I to VI is 26-28 days.
 

VII. Supplementary Topics

The following topics are frequently included in a beginning course.  
Choices will be influenced by time available, needs and interests of
students, and course objectives: Computational experience. abstract vector
spaces, linear transformations, positive definite matrices, reduction of a
symmetric matrix to a diagonal matrix by congruence, singular value
decomposition, matrix norms, Some applications, such as Markov chains,
input-output models, Leslie matrices, difference equations, differential
equations, linear programming.
 

The Use of Technology in the First Linear Algebra Course.

In addition, the LACSG wrote that "Faculty should be encouraged to utilize
technology in the first linear algebra course" where "technology" is
defined as computer or supercalculator use for homework and projects. This
will allow realistic applied problems to be solved. Suitable currently
available software packages do not require previous computer programming
experience.