MATH 206 Linear Algebra I

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MATH 206 Linear Algebra I 1. Catalog Description MATH 206 Linear Algebra I 4 units Prerequisite: MATH 143. Systems of linear equations. Matrix algebra, including inverses and determinants. Vectors, vector spaces, bases and linear transformations in real coordinate space of n dimensions. Eigenvalues, eigenvectors and diagonalization. Applications of linear algebra. Introduction to inner products and orthogonality. 4 lectures. 2. Required Background or Experience Math 143 or consent of instructor. 3. Learning Objectives The student should be able to: a. Understand the concept of matrices and their role in linear algebra and applied mathematics. b. Have a complete understanding of linear systems Ax = b, and the role of rank, subspace, linear independence, etc. in the analysis of these systems. c. Understand eigenvalues and eigenvectors of matrices and their computation. d. Know the concept of determinant and its properties. e. Understand the concepts of vector space and linear maps when the vector space is Rn. f. Understand important definitions in linear algebra and the ability to do very elementary proofs. 4. Text and References To be chosen by instructor. Suggested texts include: • Lay, David C., Linear Algebra and its Applications • Bretscher, Otto, Linear Algebra with Applications 5. Minimum Student Materials Paper, pencils, and notebook. 2019/20 6. Minimum University Facilities Classroom with ample chalkboard space for class use. Use of a computer lab is optional. 7. Content and Method a. Systems of linear equations b. Matrix and vector operations c. Properties of Rn (linear combinations, bases, spanning set, dimensions, dimension equation) d. Coordinate systems and change of basis e. Determinants f. Eigenvalues and eigenvectors g. Orthogonality h. Diagonalization of symmetric matrices i. Further topics selected by instructor 8. Methods of Assessment The primary methods of assessment are: essay examinations, quizzes and homework. Typically, there will be one or more hour-long examinations during the quarter, and a required comprehensive final examination. 2019/20.

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Real Composition Algebras by Steven Clanton
Real Composition Algebras by Steven Clanton A Thesis Submitted to the Faculty of The Wilkes Honors College in Partial Fulfillment of the Requirements for the Degree of Bachelor of Arts in Liberal Arts and Sciences with a Concentration in Mathematics Wilkes Honors College of Florida Atlantic University Jupiter, FL May 2009 Real Composition Algebras by Steven Clanton This thesis was prepared under the direction of the candidates thesis advisor, Dr. Ryan Karr, and has been approved by the members of his supervisory committee. It was sub- mitted to the faculty of The Honors College and was accepted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Liberal Arts and Sciences. SUPERVISORY COMMITTEE: Dr. Ryan Karr Dr. Eugene Belogay Dean, Wilkes Honors College Date ii Abstract Author: Steve Clanton Title: Real Composition Algebras Institution: Wilkes Honors College of Florida Atlantic University Thesis Advisor: Dr. Ryan Karr Degree: Bachelor of Arts in Liberal Arts and Sciences Concentration: Mathematics Year: 2009 According to Koecher and Remmert [Ebb91, p. 267], Gauss introduced the \com- position of quadratic forms" to study the representability of natural numbers in binary (rank 2) quadratic forms. In particular, Gauss proved that any positive defi- nite binary quadratic form can be linearly transformed into the two-squares formula 2 2 2 2 2 2 w1 + w2 = (u1 + u2)(v1 + v2). This shows not only the existence of an algebra for every form but also an isomorphism to the complex numbers for each. Hurwitz gen- eralized the \theory of composition" to arbitrary dimensions and showed that exactly four such systems exist: the real numbers, the complex numbers, the quaternions, and octonions [CS03, p.
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