Common Course Outline

Catalog Description

Course Description

This course is an in-depth look at topics such as ordinary differential equations, vector spaces, systems, linear transformations, and applications.

Date Last Revised (Month, year)

March, 2022

Outline of Major Content Areas

1. Differential Equations
   1. First and Second Order Ordinary Differential Equations (ODE)
   2. Higher Order Linear and Non-Linear ODE
   3. Real World applications with first and higher order ODE
   4. Numerical Methods for ODE
   5. Systems of Linear ODE and Dynamical Systems
   6. Laplace Transforms
   7. Power Series Methods
2. Linear Algebra
   1. Matrix Arithmetic Operations and Solutions for Systems and Homogeneous Systems
   2. Classification of Matrix Structures
   3. Determinants and their Applications
   4. Vector Space, Subspaces, Fundamental Subspaces, Inner Products, and Basis
   5. Linear Transformations and Matrix Representation
   6. Eigenvalues, Eigenvectors, Similarity, Diagonalization
   7. Gram-Schmidt Orthogonalization Process and Applications to Fourier Analysis

Learning Outcomes (General)

The student will be able to:

1. Differential Equations
   1. Use Quantitative or Directional Fields techniques to solve first, second, and higher order linear and nonlinear ordinary differential equations
   2. Solve higher order linear and nonlinear ODE
   3. Apply ODE modeling to real world applied problems including Boundary and Initial Value Problems.
   4. Apply numerical methods including Euler’s Method to ODE.
   5. Analyze systems of first order ODE and Dynamical Systems both quantitatively and qualitatively.
   6. Solve ODE and Systems of ODE via Laplace Transforms.
   7. Solve ODE by Power Series methods.
   8. Discuss and apply the existence and uniqueness theorems for differential equations.
2. Linear Algebra
   1. Perform all basic Matrix arithmetic operations such as Addition, Subtraction, Multiplication, and Row Reduction.
   2. Identify symmetric, skew-symmetric, lower triangular, upper triangular, triangular, scalar, and diagonal matrices and apply their basic properties.
   3. Solve Homogeneous and Non-Homogeneous Linear Systems by Substitutions, Elimination, and Matrix Methods.
   4. Compute and utilize Determinants and solve systems with Cramer’s Rule.
   5. Perform Matrix Factorizations including LU and QR factorizations.
   6. Verify the conditions for Vector Spaces, Spans, Linear Independence, and Basic definitions.
   7. Work the four fundamental (Row, Column, Null, Left Null).
   8. Create an orthonormal basis for a finite dimensional space by the Gram-Schmidt Algorithm.
   9. Compute Eigenvalues, Eigenvectors, and Generalized Eigenvectors.
   10. Verify a transformation is Linear and find its matrix representation with regard to a given bases.

Learning Outcomes (MnTC)

NA

Methods for Evaluation of Student Learning

Methods may include but are not limited to:

• Examinations
• Quizzes
• Homework
• Projects
• Comprehensive Final Exam

RCTC Core Outcome(s)

This course contributes to meeting the following RCTC Core Outcome(s):

Critical Thinking

Students will think systematically and explore information thoroughly before accepting or formulating a position or conclusion.

Special Information (if any)

None