MATH 2082 - Linear Algebra and Differential Equations

Credits: 5
Hours/Week: Lecture None Lab None
Course Description: This is a first course in Differential Equations including ordinary differential equations, slope fields, existence and uniqueness, boundary and initial-value problems, Laplace transforms, characteristic equations, homogeneous and nonhomogeneous equations, undetermined coefficients, power series solutions, matrix formulation of linear systems of differential equations, phase planes and stability. The companion topics from Linear Algebra include systems of linear equations and matrices, matrix operations, determinants, singular and non-singular matrices, inverse matrices, row space, column space, and null space, rank and kernel, vector spaces and subspaces, independence, bases, linear transformations, eigenvalues and eigenvectors. Use of a 3-D graphing calculator, such as a TI-Nspire CX CAS, is required. Limited use of a computer algebra system will be made.
MnTC Goals
None

Prerequisite(s): MATH 1082  with a grade of C or higher, or instructor consent.
Corequisite(s): None
Recommendation: Eligible for college-level Reading and English.

Major Content
1. Systems of Linear Equations and Matrices

2. Vector Spaces and Linear Transformations

3. First-Order Differential Equations

4. Higher-Order Differential Equations

5. Systems of Differential Equations

1. demonstrate critical and logical reasoning when solving problems.
2. solve linear systems of equations by linear algebra and matrix methods including Gaussian Elimination, Gauss-Jordan Elimination, and by matrix equation representation.
3. perform operations on matrices including addition, subtraction, multiplication, transposition, and inversion.
4. determine whether a vector is in the span of a finite collection of vectors.
5. prove or disprove that a given finite set of vectors is linearly independent.
6. create a basis for a nonzero finite-dimensional vector space and find its dimension.
7. construct bases for the row, column, and null space of a matrix, relating their dimensions to one another and to the rank and nullity of the matrix.
8. identify a vector space from the axioms.
9. prove that a non-empty subset of a vector space is a vector space.
10. compute, explain, and apply key properties and definitions related to eigenvalues and eigenvectors of a matrix.
11. solve homogeneous and nonhomogeneous first-order and higher-order differential equations by symbolic methods.
12. recognize and solve first- and second-order linear and nonlinear differential equations.
13. model real-life situations using first-order differential equations.
14. find numerical solutions of ordinary differential equations using Euler’s and Runge-Kutta methods.
15. illustrate solutions of differential equations using direction fields.
16. solve boundary and initial-value problems.
17. apply the Existence and Uniqueness theorem.
18. recognize and solve higher-order differential equations.
19. model real-life situations using higher-order differential equations.
20. solve problems using the Laplace transform.
21. apply series solutions of linear differential equations.
22. solve systems of differential equations.
23. express a dynamical system as a mathematical model.
24. communicate clearly a problem’s solution and its explanation for the intended audience in terms of the problem posed.

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