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# 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

a.  Solve linear systems

i.  Gaussian elimination

ii. Gauss-Jordan elimination

iii. Matrix equations

b.  Matrix operations

c.  Calculate determinants

d.  Singular and non-singular matrices

e.  Inverse matrices

2. Vector Spaces and Linear Transformations

a.  Definition, axioms, and properties of vector spaces

b.  Subspaces

c.  Row space and column space

d. Linear independence

e  Linear combinations of vectors

f.  Span of a set of vectors

g. Basis and dimension

h. Linear transformations

i. Rank, nullity, and kernel

j. Find Eigenvalues and Eigenvectors

3. First-Order Differential Equations

a.  Slope fields and graphical solutions

b.  Existence and uniqueness

c.  Solution methods for first-order

differential equations

d. Boundary and initial value problems

e.  Applications and modeling

f.  Numerical solutions using Euler’s and Runge-Kutta methods

4. Higher-Order Differential Equations

a. Reduction of order

b. Second- and higher-order linear differential equations

c. Characteristic polynomials

d. Non-homogeneous case

e. Undetermined coefficients or variation of parameters

f. Applications and modeling

g. Harmonic motion and oscillations

h. Laplace transforms

i. Power series solutions

5. Systems of Differential Equations

a. Linear systems of differential equations

b. Apply Eigenvalues and Eigenvectors

c. Model dynamical systems

d. Nonlinear systems of differential equations

e. Phase planes and stability of equilibrium
Learning Outcomes
At the end of this course students will be able to:

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.

Competency 1 (1-6)
None
Competency 2 (7-10)
None