Math 241 Linear Algebra (1) 
(Spring 2016)

Course Objectives

Student will be able to:

• Prove elementary statements concerning the theory of systems of linear equations. 
• Understand some applications of systems of linear equations. 
• Perform the operations of addition, scalar multiplication, and multiplication, and find the transpose and inverse of a matrix. 
• Calculate determinants using row operations, column operations, and expansion down any column and across any row. 
• Prove elementary statements concerning the theory of matrices and determinants. 
• Prove algebraic statements about vector addition, scalar multiplication, inner products, norms, orthogonal vectors, linear independence, spanning sets, subspaces, bases, and dimension for Rn and abstract vector spaces. 
• Write the relationships between A being invertible, det A, AX = 0 having a solution, the rank of A, and the rows of A being linearly independent. 
• Use the Gram-Schmidt process to orthogonalize vectors. 
• Find the kernel, range, rank, and nullity of a linear transformation. 
• Find the matrix associated with a linear transformation with respect to given bases, and understand the relationship between the operations on linear transformations and their corresponding matrices. 
• Find the change-of-basis matrix. 
• Calculate eigenvalues and their corresponding eigenspaces. 
• Determine if a matrix is diagonalizable, and if it is, diagonalize it. 
• Prove elementary facts concerning eigenvalues and eigenvectors. 

Course Description

The course begins with the study of systems of linear equations, their applications, and solutions. Matrices are studied, starting with simple matrix operations, and then inverses and determinants of matrices. Next, the focus is on the vector spaces, discussing elementary operations on vectors, linear independence, spanning sets, bases, the rank of a matrix, orthogonal bases, and the Gram-Schmidt process. Then vector spaces are studied in an abstract setting, examining the concepts of linear independence, span, bases, subspaces, and dimension. There follows a discussion of the association between linear transformations and matrices, as well as the kernel and range of a linear transformation. Finally, eigenvalues, eigen-vectors, and eigenspaces are discussed, as well as similar matrices and diagonalizable matrices

Course Outline

1. Matrices : Matrix Definitions, Operations on matrices, Inverse of a matrix, Elementary matrices, LU factorization, determinants, properties of determinants. (PDF Link)
2. Solving Linear system of equations: Gauss elimination , Gauss Jordan method, LU factorization.
3. Vector Spaces and Subspaces: Vector space, Subspace of a vector space, Linear combinations, Linear independence, Basis and dimensions.
4. Column Space, Row space and Null space of a matrix, Rank and nullity.
5. Inner product Space, Orthogonal ans orthonormal basis, Gram Schmidt orthogonalization, QR factorization.
6. Eigen value Problem, eigen values and eigen vectors, diagonalization of semisimple matrices, Cayley Hamilton theorem, Functions on matrices.
7. Linear Transformations, inverse linear transformations, Matrix representation.

Slides and Course Notes

1. Matrices : Matrix Definitions, Operations on matrices, Inverse of a matrix, Elementary matrices, LU factorization, determinants, properties of determinants.(PDF Link)
2. Solving Linear system of equations: Gauss elimination , Gauss Jordan method, LU factorization. (PDF Link)
3. Vector Spaces and Subspaces: Vector space, Subspace of a vector space, Linear combinations, Linear independence, Basis and dimensions.(PDF Link1) (PDF Link2)
4. Column Space, Row space and Null space of a matrix, Rank and nullity.(PDF Link)
5. Inner product Space, Orthogonal ans orthonormal basis, Gram Schmidt orthogonalization, QR factorization.(PDF Link)
6. Eigen value Problem, eigen values and eigen vectors, diagonalization of semisimple matrices, Cayley Hamilton theorem, Functions on matrices.(PDF Link1) (PDF Link2)
7. Linear Transformations, inverse linear transformations, Matrix representation.(PDF Link)

Materials 
link to course materials (here)
References

1- “Elementary Linear Algebra” Howard Anton and Chris Rorres, Ninth Edition, 2005[ View PDF]
2- “Elementary Linear Algebra” Stephen Andrilli and David Hecker, fourth Edition, 2010.