  
Student Learning Outcomes 
 Students will be able to use a linear system to model a problem. They will use techniques learned in class to solve the system, and interpret the result.
 Students will be able to understand the concepts of a vector space and prove facts about them.

Description  
 A first course in Linear Algebra, including systems of linear equations, matrices, linear transformations, determinants, abstract vector spaces and subspaces, eigenvalues and eigenvectors, inner product spaces and orthogonality, and selected applications of these topics.


Course Objectives  
 The student will be able to:
 Solve linear systems using various methods in Linear Algebra, and analyze the systems.
 Demonstrate an understanding of matrix operations, their properties, and various characterizations of invertible matrices including the Invertible Matrix Theorem.
 Evaluate the determinant and demonstrate an understanding of its properties.
 Demonstrate an understanding of vector spaces and subspaces, identify those spaces and understand their characterizations.
 Demonstrate an understanding of eigenvectors, eigenvalues, and their usage in many fields.
 Demonstrate an understanding of an orthogonal projection of a vector onto a subspace, and solve related problems in Linear Algebra.
 Write linear systems to model phenomena from various reallife problems, and discuss their solutions to demonstrate an understanding of applications of linear algebra.
 Prove various theorems/results involving any of the topics in (A) through (F) above using accurate mathematical language and notation to communicate arguments clearly.
 Use technology such as graphing calculators and/or computer software to assist in solving problems involving any of the topics in (A) through (G) above.

Special Facilities and/or Equipment  
  Graphing Calculator/Mathematica/Matlab
 When taught hybrid: Four lecture hours per week in facetoface contact and one hour per week using CCC Confer. Students need internet access.

Course Content (Body of knowledge)  
  Solve linear systems using various methods in Linear Algebra, and analyze the systems.
 Solutions of linear systems
 Elementary row operations and row echelon and row reduced echelon forms
 Gaussian elimination method
 Existence and uniqueness questions about a linear system
 Parametric descriptions of solution sets
 Vector equations
 Matrix equations
 Matrix transformations
 Linearly independent and dependent sets in R^n
 Linear transformation from R^n to R^m and its standard matrix
 Onetoone and onto linear transformations
 Demonstrate an understanding of matrix operations, their properties, and various characterizations of invertible matrices including the Invertible Matrix Theorem.
 Matrix operations and their properties
 Special types of matrices
 Transpose of a matrix
 Elementary matrices
 Inverse of a matrix and its properties
 Invertible Matrix Theorem
 Evaluate the determinant and demonstrate an understanding of its properties.
 Definition and properties of determinants
 Cofactor expansions
 Row operations and evaluation of determinants
 Demonstrate an understanding of vector spaces and subspaces, identify those spaces and understand their characterizations.
 Vectors in R^n space
 Abstract vector spaces and subspaces
 Special subspaces of a matrix : Null space, Column space, and Row space
 A subspace spanned by a set
 Linear transformation from a vector space to a vector space
 Kernel and range of a linear transformation
 Linearly independent and dependent sets in a general vector space
 Basis and dimension of a vector space
 Changeofcoordinates matrix
 Coordinate mapping
 Change of basis
 The rank of a matrix and the Rank Theorem
 Demonstrate an understanding of eigenvectors, eigenvalues, and their usage in many fields.
 Eigenvalue, eigenvector, and eigenspace of a matrix
 The characteristic equation
 Similar matrices and eigenvalues
 Diagonalization of a matrix
 Application to discrete dynamical systems
 Demonstrate an understanding of an orthogonal projection of a vector onto a subspace, and solve related problems in Linear Algebra.
 Inner product on R^n space
 Norm of a vector in R^n space
 Orthogonality of two vectors in R^n space
 Orthogonal complements
 Orthogonal sets and orthonormal sets
 Orthogonal basis and orthonormal basis
 Orthogonal projection of a vector onto a vector space
 Orthogonal decomposition
 GramSchmidt process
 Leastsquares solution
 General inner product space
 Orthogonal diagonalization of a symmetric matrix
 Write linear systems to model phenomena from various reallife problems, and discuss their solutions to demonstrate an understanding of applications of linear algebra.
 Linear models from realworld applications
 Analyze and interpret the solution set in given contexts
 Prove various theorems/results involving any of the topics in (A) through (F) above using accurate mathematical language and notation to communicate arguments clearly.
 Use technology such as graphing calculators and/or computer software to assist in solving problems involving any of the topics in (A) through (G) above.

Methods of Evaluation  
  Written homework and/or projects.
 Quizzes, tests
 Proctored comprehensive final examination

Representative Text(s)  
 Lay, David. Linear Algebra And Its Applications. 4th Ed. Reading, MA: AddisonWesley Publishing Company, 2012.

Disciplines  
 Mathematics


Method of Instruction  
  Lecture
 Discussion
 Cooperative learning exercises


Lab Content  
 Not applicable.


Types and/or Examples of Required Reading, Writing and Outside of Class Assignments  
  Homework Problems: Homework problems covering subject matter from text and related material ranging from 30  60 problems per week. Students will need to employ critical thinking in order to complete assignments.
 Lecture: Five hours per week of lecture covering subject matter from text and related material. Reading and study of the textbook, related materials and notes.
 Projects: Student projects covering subject matter from textbook and related materials. Projects will require students to discuss mathematical problems, write solutions in accurate mathematical language and notation and interpret mathematical solutions. Projects may require the use of a computer algebra system such as Mathematica or MATLAB.
 Worksheets: Problems and activities covering the subject matter. Such problems and activities will require students to think critically. Such worksheets may be completed both inside and/or outside of class.
