LINEAR ALGEBRA SYLLABUS.doc

LINEAR ALGEBRA SYLLABRA1. Course nameLinear Algebra2. SummaryLinear algebra is an important component of undergraduate mathematics, particularly for students majoring in the scientific, engineering, and social science disciplines. At the practical level, matrix theory and the related vector-space concepts provide a language and a powerful computational framework for posing and solving important problems. Beyond this, elementary linear algebra is a valuable introduction to mathematical abstraction and logical reasoning because the theoretical development is self-contained, consistent, and accessible to most students.3. PrerequisitesCalculus 4. Duration of the courses48 hours (16 weeks, 3 hours per week at classroom)5. TextbooksIntroduction to Linear Algebra; fifth edition; Lee W. Johnson, R. Dean Riess and Jimmy T. Arnold; Pearson Education North Asia Limited and China Machine Press.6. SyllabusChapter 1: Matrices and Systems of Linear EquationsIn this chapter we discuss systems of linear equations and methods (such as Gauss-Jordan elimination) for solving these systems. We introduce matrices as a convenient language for describing systems and the Gauss-Jordan solution method.1.1 Introduction to matrices and systems of linear equations After reading this section, you should be able to:Comprehend the concept of linear systemsUnderstand the geometric interpretations of solution setsUnderstand the concept of matricesRepresent a linear system using matrixComprehend elementary operations1.2 Echelon form and Gauss-JordanAfter reading this section, you should be able to: Understand the definition of echelon form and reduced echelon form Solving a linear system whose augmented matrix is in reduced echelon form For every matrix can be reduction to reduced echelon form1.3 Consistent systems of linear equationsAfter reading this section, you should be able to:Solving a consistent linear systemComprehend the concept and method of homogeneous systems1.4 Applications (optional)1.5 Matrix operations After reading this section, you should be able to:Comprehend the method of matrix addition and scalar multiplicationUnderstand the concept of vectors in RnDescribe the general solution using vector formComprehend the scalar product of two vectorsComprehend the matrix multiplication1.6 Algebraic properties of matrix operationsAfter reading this section, you should be able to:Comprehend the properties of matrix operationsUnderstand the transpose of a matrix and identity matrix1.7 Linear independence and nonsingular matricesAfter reading this section, you should be able to:Comprehend the concepts of linearly independent, linearly dependent and linear combinationUnderstand nonsingular matrices and singular matrices1.8 Data fitting, numerical integration, and numerical differentiation (optional)1.9 Matrix inverses and their properties After reading this section, you should be able to:Comprehend the concept of matrix inverseUsing inverses to solve systems of linear equationsCalculating the inverse using elementary row operationsComprehend the properties of matrix inverseUnderstand the ill-conditioned matricesUnderstand population dynamics and partitioned matricesChapter 2: Vectors in 2-space and 3-spaceIn this chapter we review the related concepts of physical vectors, geometric vectors, and algebraic vectors. To provide maximum geometric insight, we concentrate on vectors in two-space and three-space.2.1 Vectors in the plane After reading this section, you should be able to:Understand three types of vectorsUsing algebraic vectors to calculate the sum and difference of geometric vectorsComprehend scalar multiplicationUnderstand lengths of vectors and unit vectorsUnderstand the basic vectors i and j2.2 Vectors in space After reading this section, you should be able to: Understand the coordinate axes and rectangular coordinates for points in three spaceComprehend the distance formula and the midpoint formulaComprehend addition, difference and scalar multiplication for vectorsUnderstand parallel vectors, lengths of vectors, unit vectors, and basic unit vectors in three space2.3 The dot product and the cross product After reading this section, you should be able to:Comprehend the concept of dot productUnderstand the angle between two vectors and orthogonal vectorsComprehend the algebraic properties of the dot productExpressing u as the sum of two orthogonal vectorsComprehend the concept of cross productUsing determinants to remember the form of the cross productComprehend the algebraic properties of the cross productUnderstand the right-hand ruleUnderstand the geometric properties of the cross productComprehend the triple products2.4 Lines and planes in space After reading this section, you should be able to:Comprehend the equation of a line in the xy-planeComprehend the vector form equation and parametric equation for a line in three spaceUnderstand the planes in space and their normal vectorsComprehend the vector and scalar form equation for a plane in three spaceUsing the cross product to find a normalUnderstand the parallel planesChapter 3: The vector space RnIn this chapter we extend geometric vector concepts to n-dimensional space. For instance, we will see that lines and planes in three-space give rise to the idea of a subspace of Rn.3.1 Introduction3.2 Vector space properties of Rn After reading this section, you should be able to:Understand the concepts of vector space and subspacesComprehend the properties for vector spaceVerifying that subsets are subspaces3.3 Examples of subspaces After reading this section, you should be able to:Understand the span of a subset, the null space of a matrix, the range of a matrix and the row space of a matrix3.4 Base for subspaces After reading this section, you should be able to:Understand the concept of spanning sets and minimal spanning sets3.5 Dimension After reading this section, you should be able to:Comprehend the definition of dimensionComprehend the properties of a p-dimensional subspaceUnderstand the rank of a matrix3.6 Orthogonal bases for subspaces After reading this section, you should be able to:Understand the concept of orthogonal bases and orthonormal basesDetermining coordinatesConstructing an orthogonal basis3.7 Linear transformations from Rn to Rm After reading this section, you should be able to:Understand the concept of Linear transformations from Rn to RmComprehend the matrix of a transformationUnderstand null space and range3.8 Least-squares solutions to inconsistent systems, with applications to data fitting (optional)3.9 Theory and practice of least squares (optional)Chapter 4: The eigenvalue problemAs we shall see, the eigenvalue problem is of great practical importance in mathematics and applications. In section 4.1 we introduce the eigenvalue problem for the special case of (22) matrices; this special case can be handled using ideas developed in Chapter 1. If you have time and if you want a thorough discussion of determinants, you might want to cover Chapter 6 (determinants) before Chapter 4 (The eigenvalue problem). Chapter 4 and Chapter 6 are independent, and they are designed to be read in and order.4.1 The eigenvalue problem for (22) matrices After reading this section, you should be able to:Comprehend the concept of the eigenvalue problemCalculating the eigenvalues for (22) matricesCalculating the eigenvectors for (22) matrices4.2 Determinants and the eigenvalue problem (optional)4.3 Examples of subspaces (optional) After reading this section, you should be able to:Understand the span of a subset, the null space of a matrix, the range of a matrix and the row space of a matrix4.4 Eigenvalues and the characteristic polynomial After reading this section, you should be able to:Comprehend the concept of the characteristic polynomialComputational considerations4.5 Eigenvectors and eigenspaces After reading this section, you should be able to:Understand the concept of the eigenspaces and geometric multiplicity4.6 Complex eigenvalues and eigenvectors (optional)4.7 Similarity transformations and diagonalization After reading this section, you should be able to:Understand the concept of similarity and diagonalizationUnderstand the concept of orthogonal matricesDiagonalize symmetric matrices4.8 Difference equations; markov chains; systems of differential equations (optional)Chapter 5: Vector spaces and linear transformationsIn this chapter, using Rn as a model, we further extend the idea of a vector to include objects such as matrices, polynomials, functions, infinite sequences, and so forth.5.1 Introduction5.2 Vector spaces After reading this section, you should be able to:Understand the concepts of vector spaceComprehend the properties for vector space5.3 Subspaces After reading this section, you should be able to:Understand the spanning sets5.4 Linear independence, bases, and coordinates After reading this section, you should be able to:Understand the concept of linear independence and linear dependenceUnderstand the concept of vector-space bases5.5 Dimension After reading this section, you should be able to:Understand the properties of a p-dimensional vector space5.6 Inner-product spaces, orthogonal bases, and projections (optional)5.7 Linear transformations After reading this section, you should be able to:Understand the concept of Linear transformationsComprehend the properties of a linear transformations5.8 Operations with linear transformations After reading this section, you should be able to:Understand the concept of invertible transformationsUnderstand isomorphic vector spaces5.9 Matrix representations for linear transformations After reading this section, you should be able to:Comprehend the matrix of a transformationUnderstand the representation theoremComprehend the algebraic properties of transformations5.10 Change of basis and diagonalization After reading this section, you should be able to:Diagonalizable transformationUnderstand the transition matrixComprehend matrix representation and change of basisChapter 6: DeterminantsIn this chapter we introduce the idea of the determinant of a square matrix. We also investigate some of the properties of the determinant. For example, a square matrix is singular if an