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May 31, 2023
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# MATH 205 Linear Algebra (5 credits)

Distribution Area Fulfilled Natural Sciences; Quantitative and Symbolic Reasoning; General Transfer Elective
Prerequisite MATH& 153  or MATH& 163  with a grade of 2.0 or higher (MATH 224  or MATH& 264  recommended) or instructor permission.

Course Description
Applications and techniques of Linear Algebra, including solving systems of equations, vector spaces, matrix operations, linear transformations, eigenvalues, eigenvectors, and characteristic polynomials. Introduction to appropriate technology and elementary proofs.

Student Outcomes
The student should be able to:

Gaussian Elimination component
1. Row reduce a matrix to reduced row echelon form by hand.
2. Use Gaussian elimination to find all solutions to a system of linear algebraic equations.
3. Use systems of equations to effectively model real world problems from engineering and the sciences and interpret their solutions in the context of the problem.

Matrix component
4. Add, subtract, multiply, and re-scale matrices without technology.
5. Compute the inverse of a matrix using Gaussian elimination without technology.
6. Use the inverse of a matrix to solve a system of equations without technology.
7. Compute the transpose of a matrix without technology.
8. Utilize the properties of the determinant in solving problems in linear algebra.
9. Calculate determinants without technology.

Vector Spaces component
10. Define and understand the concept of a vector space.
11. Prove whether or not a given set with given operations is a vector space, including Euclidean spaces with standard and non-standard operations, polynomial spaces, function spaces, matrix spaces.
12. Determine whether or not a subset of a given vector space is a subspace.
13. Compute the span of a given subset of a vector space.
14. Determine whether or not a given subset of a vector space is linearly independent.
15. Define and understand the concept of a basis for a vector space.
16. Determine whether or not a given subset is a basis, including the standard bases for Euclidean space, polynomial spaces, and matrix spaces.
17. Determine the dimension of Euclidean, polynomial, and matrix spaces.
18. Compute the rank and nullspace of a matrix, including finding appropriate bases.
19. Compute the coordinates of an element of a vector space relative to a given basis.
20. Compute the transition matrix for coordinates relative to two bases.

Linear Transformations component
21. State the definition of a linear transformation and determine whether a given transformation is linear or not.
22. View examples from calculus as linear transformations, including preparations for differential equations.
23. Compute the kernel and range of a linear transformation, including finding appropriate bases, including for linear transformations not given by a matrix.
24. Find the matrix representation of a linear transformation relative to standard and non-standard bases.
25. Construct matrix representations of geometric transformations such as reflections, dilations, contractions, and reflections in Euclidean space.
Eigenvalues component
26. Visualize the geometric consequences of the eigenvalues and eigenvectors of a matrix.
27. Compute the eigenvalues of a matrix via the characteristic polynomial.
28. Compute the eigenvectors of a matrix via the nullspace.
29. Diagonalize a matrix.
30. Apply diagonalization to solve a variety of problems from mathematics, the sciences, and engineering, including the Fibonacci numbers, stochastic processes, and Markov processes.
General Content
31. Write clear, correct, and complete solutions to mathematical problems utilizing proper mathematical notation and appropriate language.
32. Compute basic examples and concepts by hand.
33. Write clear, coherent, and correct mathematical proofs at a basis level, including construction of counter examples and proof by contradiction.
34. Utilize computer algebra and graphical systems to solve problems, understand concepts, and model physical problems, as appropriate.