Math 130, Linear Algebra
Fall 2013 Prof. D. Joyce, BP 322, 793-7421 Department of Mathematics and Computer Science Clark University |

William Rowan Hamilton | |||
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William Rowan Hamilton (1805–1865)
The Pantheon in the Library—The Sculpture Busts of Long Room Trinity College, Dublin Sculptor, John Henry Foley (1818–1874) Photographer, Arran Q Henderson |

[This course page is obsolete.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.

General information

**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.

See also Clark’s*Academic Catalog*.**Prerequisites.**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.**Textbook.**Linear Algebra, 4th edition, by Friedberg, Insel, and Spence, published by Pearson, 2003. ISBN-10: 0130084514, ISBN-13: 9780130084514For details, see http://www.pearsonhighered.com/educator/product/Linear-Algebra/9780130084514.page

You may find used books are less expensive than new ones.

Syllabus

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.

- Vectors and vector spaces.
- Vectors in
. Their addition, subtraction, and multiplication by scalars (i.e. real numbers). Graphical interpretation of these vector operations**R**^{n} - 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
. Matrices, row vectors, column vectors, polynomials, infinite sequences**R**^{n} - Fields defined abstractly. Axioms and theorems that follow from them.
- The complex field
and complex vector spaces**C**.**C**^{n} - Finite fields and their vector spaces.
- Subspaces. Lines in the plane
**R**^{2}, lines and planes in**R**^{3}. 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.

- Vectors in
- 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*:→**R**^{n}. Linear operators on**R**^{m}**R**^{2}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. Inverse matrices
- The change of coordinate matrix between two different bases of a vector space. Similar matrices.
- Dual Spaces.
- [A matrix representation for complex numbers, and another for quaternions. Historical note on quaternions.]

- Linear transformations. The definition of a linear transformation
- 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)

- Determinants.
- 2x2 Determinants of Order 2. Multilinearity. Inverse of a 2x2 matrix. Signed area of a plane parallelogram, area of a triangle.
*n*x*n*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
**R**^{3}]

- 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 "perpendicular")
- Unit vectors and standard unit vectors in
**R**^{n} - Orthonormal basis

To be filled in as the course progresses.

- Vectors
- Fields
- 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 linear algebra
- A little bit about sets
- Simultaneous systems of linear equations.

The Chinese method of elimination at http://aleph0.clarku.edu/~djoyce/ma105/simultaneous.html - 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 source file
- Linear combinations and basis
- Span, and independence
- The replacement theorem and dimension
- Where dimension doesn’t work
- Isomorphisms
- Linear transformations, matrices, and linear operators
- First test Oct 7 on vectors spaces (through the chapter 1.5). Answers
- Some linear transformations of the plane
**R**^{2} - An applet to draw fractals using a Lindenmayer L-system
- Coordinates
- 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.

Answers

Extra credit problems for second test. Answers - Eigenvalues, eigenvectors, and eigenspaces of linear operators
- Rotations and complex eigenvalues
- Diagonalizable operators and matrices

Answers - Norm and inner products in
**R**^{n} - Applications of inner products in
**R**^{n} - Norm and inner products in
**C**^{n}and abstract inner product spaces - Cauchy’s inequality
- Quaternions
- Cross products

- Tests from last year.
- 2012 first midterm, answers
- 2012 second midterm, answers
- 2012 final, answers

- Math 120, Calculus I
- Math 121, Calculus II
- Math 131, Multivariate Calculus
- Math 217, Probablility and Statistics
- Math 218, Mathamatical Statistics
- Math 225, Modern Algebra

Pages on the web that you may find interesting

- Dave’s Short Trig Course, and Dave’s Short Course on Complex Numbers.
- Articles by J J O’Connor and E F Robertson on the history of Abstract linear spaces and on the history of Matrices and determinants at The MacTutor History of Mathematics archive.
- Dave Rusin’s
*The Mathematical Atlas*describes the subject of linear algebra and its subfields, at Northern Illinois University. - Lagrange Interpolation Demo displays a function (
*x,y*)(t) which has specified values at*t*= 1, 2,...,*n*. Applet by Mark Hoefer.

This page is located on the web at

http://aleph0.clarku.edu/~djoyce/ma130/