**Mel S. Sabella
E. F. Redish**

*Student Understanding of Topics in Linear Algebra*

**Introduction**

A number of studies concerning student understanding of linear algebra at the university level have been performed in recent years. The increasing demand for student understanding of linear algebra and the growing concern that the present linear algebra course does not adequately meet the needs of the students it serves, prompts further study and curriculum development.

Five papers are discussed in this review, each presenting objectives and recommendations for courses in linear algebra. The study by Wang discusses the philosophy, assignments, and student feedback for a course designed for graduate students in chemical engineering at the University of Tennessee. The other papers, by Carlson, Harel, and Tucker, are more broad based, discussing the introduction to the linear algebra course at the university level.

**Student Difficulties**

Most of the papers state that students usually master the algorithmic skills involved in linear algebra, but lack a conceptual understanding of the subject and how to apply linear algebra concepts to physical systems. Carlson states that solving systems of linear equations and calculating products of matrices is easy for the students but when they get to subspaces, spanning, and linear independence students become confused and disoriented. Carlson provides the following reasons why certain topics in linear algebra are so difficult:

- Linear algebra is taught far earlier and to less sophisticated students than before.
- These difficult topics are concepts, not computational algorithms. Almost all mathematical experience of today's American sophomores has been computational.
- Different algorithms are required to work with these ideas in different settings. Exercises in which students determine these various procedures for themselves are generally not assigned.
- These concepts are introduced without substantial connection
with students' prior experience and without significant examples or applications
(Carlson
_{2}30).

Harel explains that students also have trouble with basic notations and states that the reason for this is that the abstract concepts come too quickly without a firm intuitive base (Harel 139). He also cites that textbooks in linear algebra are written based on the assumptions "that students recognize models and solve problems by translating them to isomorphic but abstract structures; and that they can apply the principles of abstract setting to solve problems. [He states that] these assumptions were found to be invalid . . . with many beginning university students (Harel 140)."

Wang sees graduate students exhibiting some of the same difficulties. Chemical engineering graduate students at the University of Tennessee are required to take a course entitled "Application of Numeric Linear Algebra in Systems and Control Engineering." Wang notices that students usually enter this advanced linear algebra class knowing only how to do matrix addition, subtraction, multiplication, and finding the determinant and inverse of up to 3X3 matrices. Some know a little about basis vectors and they have some notions about linear independence of vectors (Wang 236).

**Implications for Curriculum Development**

Since course content is dependent on the needs of a number
of disciplines for a linear algebra course, the math department must maintain
a constant liaison with these departments in order to monitor their needs
in higher level mathematics. Otherwise, the course will usually not serve
the needs of the science and engineering students. The first course in
linear algebra is a service course for a wide variety of disciplines such
as computer science, electrical engineering, other engineering fields such
as aerospace engineering and systems engineering, physics, economics, statistics,
and operations research. After taking the linear algebra course, students
often come away knowing how to perform certain algorithms but they have
not acquired the intuition relating knowledge of the mathematics to selection
of the method for analysis, design, and control of physical systems (Wang
237). Since students in these various disciplines often only take one
course in linear algebra the course must cover essential materials as well
as materials needed most by the majority of the students (Carlson_{1}
42). Carlson has indicated that representatives from various disciplines
desire a "solid and intellectually challenging course, with careful
definitions and statements of theorems, and proofs that show relationships
between various concepts and enhance understanding (Carlson_{1}
42)." Carlson states that based on the relatively new role of
linear algebra as a tool for industrial scientists the course should de-emphasize
abstraction and put more emphasis on problem solving and motivating applications,
although preserving the current level of rigor and theorem proving. The
motivational applications must also be well thought out. Harel points out
that certain applications such as those found in analytic geometry and
vector arithmetic do not enhance student appreciation and understanding
of the abstract content of linear algebra. The reason for this lack of
appreciation and understanding is that students do not see the necessity
of dealing with abstract concepts for treating a limited scope of situations;
and constructions of processes are generally not dealt with. In his paper,
entitled "Learning and Teaching Linear Algebra: Difficulties and an
Alternative Approach to Visualizing Concepts and Procedures," he stresses
a program built upon the idea of gradually teaching students the abstraction
process in three phases. These phases are visualizing concepts and processes,
representation and the establishment of the dimension of R_{n},
and abstract vector spaces.

In January 1990 The Curriculum Study Group was formed
to work on curricular and teaching reform in linear algebra. The group
consists of David Carlson (San Diego State University), Charles R. Johnson
(College of William and Mary), David C. Lay (University of Maryland), and
Duane Porter (University of Wyoming). The Curriculum Study Group seems
to prefer a less abstract course stating that the new role as a tool for
the industrial scientist implies a change of focus "from an abstract,
inward-looking course to a more practical matrix-oriented course that meets
the needs of . . . mathematics students . . . [as well as] students of
the various client disciplines (Carlson_{1}
42)." They state that the course content should also be taught
based on considerations of technology. In August of 1990 The Curriculum
Study Group wrote a suggested core syllabus which is included in the paper
by Carlson et al. and in the Appendix of this review.

Wang suggests emphasis be placed on intuitive understanding and geometric visualization and interpretation of the key theorems of linear algebra. Students should learn the whys of doing certain decompositions and manipulations and should be able to visualize the algorithms in 3-D space (Wang 241). Wang hopes that this emphasis will give the students the ability to "examine a [physical] system and by using fundamental linear algebra concepts, . . . extract physical information from it (Wang 237)."

**Conclusion**

All three articles, Wang, Carlson et al., and Harel, outline a different course syllabus, each of which seems to cater to different disciplines. Most of the studies emphasize the need of students to obtain physical information from the mathematics. An NSF-sponsored workshop on college linear algebra in August 1990 recommended giving more attention to matrix algebra and its applications, while endorsing the current level of theory and rigor. A report of the CUPM Subcommittee on the mathematics major finds merit in placing more emphasis on matrix methods as long as vector space theory is covered in a subsequent upper-division course. In general, linear algebra courses offered outside the mathematics department concentrate on matrix methods. Based on the amount of time provided for the linear algebra course and the nature of the student populations one must find a middle ground blending vector spaces and matrix methods (Tucker 8).

**References**

Carlson_{1},
David, (1993). The Linear Algebra Curriculum Study Group Recommendations
for the First Course in Linear Algebra, *College Mathematics Journal*,
**12** (1) 41-46.

Carlson_{2},
David, (1993). Teaching Linear Algebra: Must the Fog Always Roll In?, *College
Mathematics Journal*, **12** (1) 29-40.

Harel, Guershon,
(1989). Learning and Teaching Linear Algebra: Difficulties and an Alternative
Approach to Visualizing Concepts and Processes, *Focus on Learning Problems
in Mathematics*, **11** (2) 139-140.

Tucker,
Alan, (1993). The growing Importance of Linear Algebra in Undergraduate
Mathematics*, College Mathematics Journal*, **12** (24) 3-9.

Wang,
Tse-Wei, (1989). A Course on Applied Linear Algebra, *Chemical Engineering
Education*, **23** (4) 236-241.

**Appendix
**Core Syllabus from The Curriculum Study Group

**Prerequisite:**

Mathematical maturity associated with completion of two semesters of calculus

**Goals:**

Mastery of provided core topics

Increased problem solving capability

I. Matrix Addition and Multiplication

- matrix addition
- scalar multiplication
- matrix multiplication
- transposition
- associativity of matrix multiplication
- operations with partitioned matrices

II. Systems of Linear Equations

- Gaussian elimination
- elementary matrices
- echelon and reduced row echelon form
- existence and uniqueness of solutions
- matrix inverses
- row reduction interpreted as LU-factorization

III. Determinants

- cofactor expansion
- determinants and row operations
- det
*AB*= det*A*det*B* - Cramer's Rule

IV. Properties of R^{n}

- linear combinations: linear dependence and independence
- bases of R
^{n} - subspaces of R
^{n}: spanning set, basis, dimension, row space and column space, null space - matrices of linear transforms
- rank: row rank=column rank, products, connections with invertible submatrices
- systems of equations revisited: solution theory, rank+nullity=number of columns
- inner product: length and orthogonality, orthogonal /orthonormal sets and bases, orthogonal matrices

V. Eigenvalues and Eigenvectors

- the equation Ax=lx
- characteristic polynomial and identification of some of its coefficients (trace and determinant), algebraic multiplicity of eigenvalues
- eigenspaces, geometric multiplicity
- similarity: distinct eigenvalues and diagonalization
- symmetric matrices: orthogonal diagonalization, quadratic forms

VI. More on Orthogonality

- orthogonal projection onto a subspace: Gram-Schmidt orthogonalization and interpretation as a QR factorization
- the least square solutions of inconsistent linear systems, with applications to data-fitting

VII. Supplementary Topics

- computational experience
- abstract vector spaces
- linear transforms
- positive definite matrices
- reduction of a symmetric matrix by congruence
- singular value decomposition
- matrix norms

*The William and Mary Workshop Panel*

*Glenn Adamson, Ottawa University**Paul Bengtson, Casper College**James Bunch, University of California, San Diego**David Carlson, San Diego State University**Jane Day, San Jose State University**Guershon Harel, Purdue University**Roger Horn, John Hopkins University*

University of Maryland | Physics Department | PERG UMD | Project Links |
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*Please address questions and comments to Mel Sabella.
email: msabella@delphi.umd.edu*