[This course page is obsolete. I'll update it next time I teach the course.]
Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma130/, so you can readily access it.
- Course description.
Math 130 is a requirement for mathematics and physics majors, and it’s highly
recommended for majors in other sciences especially including computer-science
majors. Topics include systems of linear
equations and their solutions, matrices and matrix algebra, inverse matrices;
determinants and permutations; real n-dimensional vector spaces, abstract vector
spaces and their axioms, linear transformations; inner products (dot products),
orthogonality, cross products, and their geometric applications; subspaces,
linear independence, bases for vector spaces, dimension, matrix rank;
eigenvectors, eigenvalues, matrix diagonalization. Some applications of linear
algebra will be discussed, such as computer graphics, Kirchoff’s laws,
linear regression (least squares), Fourier series, or differential equations.
Clark’s Academic Catalog.
The prerequisite for the course is one year of college calculus, others by permission only.
- Course Hours. MWF 10:00–10:50. BioPhysics room 316.
- Office hours. MWF 1:00-2:30. BioPhysics room 322. Also, just stop by anytime I’m in, which is most every day.
- Assignments & tests.
There will be numerous short assignments, mostly from the text, occasional quizzes,
two tests during the semester, and a two-hour final exam during finals week in December.
The two tests during the semester are yet to be scheduled.
- Course grade. The course grade will be determined as follows:
2/9 assignments and quizzes,
2/9 each of the two midterms, and
1/3 for the final exam.
- Matlab. There are several different symbolic mathematics programs. We’ll use the one called Matlab. A couple of others you may have heard of are Maple and Mathematica. They can be used to perform various mathematical computations. You’ll need to know how to do these computations and perform small computations by hand, but for large computations, it helps to have a program do them to save time and reduce mistakes.
- Course goals.
- To provide students with a good understanding of the concepts and methods of linear
algebra, described in detail in the syllabus.
- To help the students develop the ability to solve problems using linear algebra.
- To connect linear algebra to other fields both within and without mathematics.
- To develop abstract and critical reasoning by studying logical proofs and the
axiomatic method as applied to linear algebra.
- Course objectives.
Students will be able to apply the concepts and methods described in the syllabus, they will be able to solve problems using linear algebra, they will know a number of applications of linear algebra, and they will be able to follow complex logical arguments and develop modest logical arguments. The text and class discussion will introduce the concepts, methods, applications, and logical arguments; students will practice them and solve problems on daily assignments, and they will be tested on quizzes, midterms, and the final.
Linear Algebra, 4th edition, by Friedberg, Insel, and Spence, published by Pearson, 2003. ISBN-10: 0130084514, ISBN-13: 9780130084514
For details, see
You may find used books are less expensive than new ones.
We won’t cover all of the topics listed below at the same depth. Some topics are fundamental and we’ll cover them in detail; others indicate further directions of study in linear algebra and we’ll treat them as surveys. Besides those topics listed below, we will discuss some applications of linear algebra to other parts of mathematics and statistics and to physical and social sciences.
Class notes, quizzes, tests, homework assignments
- Vectors and vector spaces.
- Vectors in Rn.
Their addition, subtraction, and multiplication
by scalars (i.e. real numbers). Graphical interpretation of these vector operations
- Topics we’ll discuss at the end of the semester when we discuss inner product
spaces: norm of a vector (also called length), unit vectors,
inner products of vectors (also called dot products).
- Real vector spaces defined abstractly. Axioms and theorems that follow from them.
- Examples of vector spaces besides Rn. Matrices, row
vectors, column vectors, polynomials, infinite sequences
- Fields defined abstractly. Axioms and theorems that follow from them.
- The complex field C and complex vector spaces
- Finite fields and their vector spaces.
- Subspaces. Lines in the plane R2,
lines and planes in R3. The 0 subspace.
- Systems of simultaneous linear equations and their solutions. Row reduction.
Solving them with Matlab.
- Linear combinations of vectors, the span of a set of vectors.
Geometric interpretation of the span.
Span and linear combination problems in Matlab
- Linear dependence and independence.
Geometric interpretation of dependence and independence
Testing for linear independence in Matlab.
- Bases and dimension. The dimension of subspaces.
Finite-dimensional versus infinite-dimensional spaces.
- Linear Transformations and Matrices.
- Linear transformations. The definition of a linear transformation
L: V → W from the domain space V to
the codomain space W. When V = W, L is
also called a linear operator on V.
- Examples L:
Rn → Rm.
Linear operators on R2 including
rotations and reflections, dilations and contractions, shear transformations,
projections, the identity and zero transformations
- The null space (kernel) and the range (image) of a transformation, and their
dimensions, the nullity and rank of the transformation
- The dimension theorem: the rand plus nullity equals the dimension of the domain
- Matrix representation of a linear transformation between finite dimensional
vector spaces with specified bases
- Operations on linear transformations V → W. The
vector space of all linear transformations V → W.
Composition of linear transformations
- Corresponding matrix operations, in particular, matrix multiplication
corresponds to composition of linear transformations. Powers of square matrices.
Matrix operations in Matlab
- Invertibility and isomorphisms. Invariance of dimension under isomorphism.
- The change of coordinate matrix between two different bases of a vector space.
- Dual Spaces.
- [A matrix representation for complex numbers, and another for quaternions.
Historical note on quaternions.]
- Elementary matrix operations and systems of simultaneous linear equations.
- Elementary row operations and elementary matrices.
- The rank of a matrix (row rank) and of its dual (its column rank).
- An algorithm for inverting a matrix. Matrix inversion in Matlab
- Systems of linear equations in terms of matrices.
Coefficient matrix and augmented matrix.
Homogeneous and nonhomogeneous equations.
Solution space, consistency and inconsistency of systems.
- Reduced row-echelon form, the method of elimination (sometimes called
Gaussian elimination or Gauss-Jordan reduction)
- 2x2 Determinants of Order 2. Multilinearity. Inverse of a 2x2 matrix.
Signed area of a plane parallelogram, area of a triangle.
- nxn determinants. Cofactor expansion
- Computing determinants in Matlab
- Properties of determinants. Transposition, effect of elementary row
operations, multilinearity. Determinants of products, inverses, and transposes.
Cramer’s rule for solving n equations in n unknowns.
- Signed volume of a parallelepiped in 3-space
- [Optional topic: permutations and inversions of permutations; even and odd permutations]
- [Optional topic: cross products in R3]
- Eigenvalues and eigenvectors of linear operators
- An eigenspace of a linear operator is a subspace in which the operator acts
as multiplication by a constant, called the eigenvalue (also called the
characteristic value). The vectors in the
eigenspace are alled eigenvectors for that eigenvalue.
- Geometric interpretation of eigenvectors and eigenvalues.
Fixed points and the 1-eigenspace.
Projections and their 0-eigenspace.
Reflections have a –1-eigenspace.
- Diagonalization question.
- Characteristic polynomial.
- Complex eigenvalues and rotations.
- An algorithm for computing eigenvalues and eigenvectors
- Inner product spaces
- Inner products for real and complex vector spaces (for real vector spaces, inner
products are also called dot products or scalar products) and norms (also called
lengths or absolute values). Inner product spaces. Vectors in Matlab.
- The triangle inequality and the Cauchy-Schwarz inequality, other properties
of inner products
- The angle between two vectors
- Orthogonality of vectors ("orthogonal" and "normal" are other words for
- Unit vectors and standard unit vectors in Rn
- Orthonormal basis
To be filled in as the course progresses.
Some old linear algebra tests
Dave’s Short Course on Complex Numbers
- Writing Proofs: structures of theorems and proofs,
synthetic and analytic proofs, logical symbols, and well-written proofs
- Vector Spaces, their axiomatic definition
- Properties of vector spaces that follow from the axioms
- Subspaces of vector spaces
- Linear differential equations, an application of
- A little bit about sets
- Simultaneous systems of linear equations.
The Chinese method of elimination at
- Notes on Matlab
- Systems of linear equations, elementary row operations,
and reduced echelon form...
- Quiz 1. Monday, Sept. 23. Answers
- Brief introduction to Latex and its
- Linear combinations and basis
- Span, and independence
- The replacement theorem and dimension
- Where dimension doesn’t work
- Linear transformations, matrices, and linear operators
- First test Oct 7 on vectors spaces
(through the chapter 1.5).
- Some linear transformations of the plane
- An applet to draw fractals using a Lindenmayer L-system
- Composition and categories, composition of linear transformations and multiplication of matrices
- Algebra of linear transformations and matrices
- Quiz 2. Monday, Oct 21. On section 1.6.
- Kernel, image, nullity, and rank
- Inverses of linear transformations and matrices
- Change of coordinates
- Elementary row operations and elementary matrices
- Rank and nullity of matrices
- Introduction to determinants, 2x2 and 3x3 determinants, areas of triangles and parallelagrams in the plane, volumes of parallelepipeds, Jacobians
- Permutations and determinants
- Characterizing properties and constructions of determinants, cofactors, diagonal and triangular matrices
- More properties of determinants, an algorithm for evaluating determinants, determinants of products, inverses, and transposes, Cramer’s rule
- Second test. Wednesday, Nov 13. Linear transformations, matrices, elementary operations, rank, nullity. Topics up through assignment 9 will be covered.
Extra credit problems for second test.
- Eigenvalues, eigenvectors, and eigenspaces of linear operators
- Rotations and complex eigenvalues
- Diagonalizable operators and matrices
- Norm and inner products in Rn
- Applications of inner products in Rn
- Norm and inner products in Cn
and abstract inner product spaces
- Cauchy’s inequality
- Cross products
Web pages for related courses
Pages on the web that you may find interesting
This page is located on the web at
David E. Joyce,