Linear Algebra (MATH-3215)

Linear algebra is the study of linear systems of equations, vector spaces, and linear transformations. Solving systems of linear equations is a basic tool of many mathematical procedures used for solving problems in science and engineering.

Prerequisite

To succeed in this course you will need to be comfortable with vectors, matrices, and three-dimensional coordinate systems.

Learning Outcomes

At the end of this course the student will be able to:

Solve linear systems of equations
Comprehend vector spaces, subspaces and inner product spaces
Understand fundamental properties of matrices including determinants, inverse matrices, matrix factorizations, eigenvalues, orthogonality and diagonalization
Have an insight into the applicability of linear algebra
Critically analyze and construct mathematical arguments that relate to the study of introductory linear algebra.

Contents

Introduction to Vectors: Vectors and Linear Combinations, Lengths and Dot Products, Matrices.
Solving Linear Equations: Vectors and Linear Equations, the Idea of Elimination, Elimination Using Matrices, Rules for Matrix Operations, Inverse Matrices.
Elimination = Factorization; A = LU, Transposes and Permutations.
Vector Spaces and Subspaces: Spaces of Vectors, The Null space of A: Solving Ax = 0, The Rank and the Row Reduced Form, the Complete Solution to Ax = B, Independence, Basis and Dimension, Dimensions of the Four Subspaces.
Orthogonally: Orthogonally of the Four Subspaces, Projections, Least Squares Approximations, Orthogonal Bases and Gram-Schmidt.
Determinants: The Properties of Determinants, Permutations and Cofactors, Cramer's Rule, Inverses, and Volumes.
Eigenvalues and Eigenvectors: Introduction to Eigenvalues, Diagonalizing a Matrix, Applications to Differential Equations, Symmetric Matrices, Positive Definite Matrices, Similar Matrices, Singular Value Decomposition (SVD).
Applications: Matrices in Engineering, Graphs and Networks, Markov Matrices, Population, and Economics; Linear Programming, Fourier series: Linear Algebra for Functions, Linear Algebra for Statistics and Probability, Computer Graphics.
Numerical Linear Algebra: Gaussian Elimination in Practice, Norms and Condition Numbers, Iterative Methods for Linear Algebra.
Complex Vectors and Matrices: Complex Numbers, Hermitian and Unitary Matrices, Matrix Factorizations.

Textbook(s)

Introduction to Linear Algebra by Gilbert Strang, Wellesley Cambridge Press; 4th Edition (February 10, 2009).

Reference Material

Elementary Linear Algebra with Applications by Bernard Kolman, David Hill, 9th Edition, Prentice Hall PTR, 2007.
Linear Algebra And Its Applications by Gilbert Strang, Strang, Brett Coonley, Andrew Bulman-Fleming, 4th Edition, 2005.
Elementary Linear Algebra: Applications Version by Howard Anton, Chris Rorres, 9th Edition, Wiley, 2005.
Linear Algebra and Its Applications by David C. Lay, 2nd Edition, Addison-Wesley, 2000.
Linear Algebra by Harold M. Edwards, Birkhäuser; 1st Edition, 2004.
Linear Algebra: A Modern Introduction by David Poole by Brooks Cole, 3rd Edition.

Suggested Books

Linear Algebra by David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron.
Linear Algebra and Its Applications by David C Lay and Steven R Lay.
Schaum’s Outline of Theory and Problem of Linear Algebra. Seymour Lipschutz. Mc-Graw Hill.

ASSESSMENT CRITERIA

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam: 30

Final-Term Exam: 50