# MATH 285 - Linear Algebra and Differential Equations

The following are the Student Learning Outcomes (SLOs) and Course Measurable Objectives (CMOs) for MATH 285. A Student Learning Outcome is a measurable outcome statement about what a student will think, know, or be able to do as a result of an educational experience. Course Measurable Objectives focus more on course content, and can be considered to be smaller pieces that build up to the SLOs.

**Student Learning Outcomes (SLOs)**

- Students can solve non-homogeneous linear differential equations of any order using a variety of methods.
- Students can formulate and solve differential equations which model real-world phenomena.
- Students can prove and apply facts regarding vector spaces, subspaces, linear independence, bases, and orthogonality.
- Students can diagonalize square matrices and apply these results to the solutions of linear systems of differential equations.
- Students can solve linear differential equations using power series.

**Course Measurable Objectives (CMOs)**

- Verify solutions to ordinary differential equations by substitution.
- Identify the type of a given differential equation and select and apply the appropriate analytical technique for finding the solutioin of first and higher order ordinary differential equations.
- Apply the existence and uniqueness theorems for ordinary differential equations.
- Set up and solve differential equations for the following applications: simple and logistic population growth model, simple electric circuits, mixing, orthogonal trajectories. Plot slope fields and numerically solve 1st order differential equations using Euler's and Runga Kutta methods.
- Demonstrate the operations of matrix algebra, row operations for linear systems, and the methods of Gaussian Elimination and matrix inversion for solving linear systems.
- Evaluate determinants using cofactors and row operations. Demonstrate the properties of determinants and matrix inversion using cofactors.
- Use bases and orthonormal bases to solve problems in linear algebra. Find the dimension of spaces such as those associated with matrices and linear transformations.
- Prove basic results in linear algebra using appropriate proof-writing techniques such as linear independence of vectors; propoerties of subspaces; linearity, injectivity and surjectivity of functions; and properties of eigenvectors and eigenvalues.
- Solve problems pertaining to the definitions of vector space, subspace, span, linear dependence and independence, basis and dimension, row and column space and inner product space. Demonstrate the use of the Gram-Schmidt process for orthogonalization.
- Solve problems pertaining to the definitions of linear transformation, kernel and range. Compute eigenvalues and eigenvectors. Diagonalize a square matrix, with the special case of orthogonal diagonalization of symmetric matrices. Demonstrate matrix representation of a linear transformation, change of bases.
- Solve linear differential equations of order n with constant coefficients (homogeneous or non-homogeneous), the methods of undetermined coefficients and variation of parameters with applications to RLC circuits or mass spring systems.
- Express a linear system of differential equations in vector form, then solve the system using eigenvalues and eigenvectors. Analyze non-linear systems numerically, including phase-plane analysis, using a computer algebra system.
- Apply the Laplace Transform and its inverse, using the basic rules of the Laplace Transform, along with the 1st Shifting Theorem. Solve linear differential equations with constant coefficients using the Laplace Transform.
- Solve ordinary differential equations using power series.