Math210 Syllabus
Math 210
Linear Algebra
Course Description
A first course in vectors, matrices, vector spaces, and linear transformations. The ideas in this course
serve as an introduction to more abstract mathematics courses at the junior-senior level, and also covers
many useful applications outside mathematics. Topics include: vectors, operations on matrices, inverse
of a matrix, solution of systems of linear equations, rank of matrix, vector spaces and subspaces, linear
dependence and independence, basis and dimension, linear transformation, sums, composites, inverses
of linear transformations, range and kernel of a linear transformation, determinants, eigenvalues and
eigenvectors, orthogonality and inner product spaces, and real quadratic forms.
Illinois Articulation Initiative (IAI) number: N/A
Credit and Contact Hours:
Lecture 3
Lab 0
Credit Hours 3
Prerequisites: Minimum grade “C” in MATH 172 or equivalent. Students may enroll concurrently in
MATH 172.
Books, Supplies, and Supplementary Materials
A. Required Textbooks
Cengage Unlimited Subscription. WebAssign will be used for online coursework
(homework, quizzes, tests, etc.) and can be accessed by logging into
iCampus/Canvas and selecting this course. If you are comfortable reading the
textbook on the computer, you may use the eText alone. There is no need to
purchase a physical textbook for this course; the Cengage Unlimited
Subscription for the eText and WebAssign was included in your course fees.
Registration instructions are posted in our iCampus/Canvas site.
B. Other Required Materials
N/A
C. Methods of Instruction:
Lecture, Hybrid, or Online
General Education Student Learning Outcome
1. Quantitative Literacy: Students possess the ability to reason and solve quantitative problems from
an array of contexts.
Course Learning Outcomes (CLOs)
1. Solve systems of linear equations using a variety of techniques
2. Explain operations on matrices, invertibility, elementary matrices
3. Demonstrate properties of determinants, including row reduction, to evaluate determinants
Math210 Syllabus
4. Identify the dimensions of a variety of vector spaces
5. Determine matrix representations of linear transformations and linear operators
6. Apply the Gram-Schmidt orthonormalization process to find an orthonormal basis for a given
basis, subspace, or inner product space
7. Calculate eigenvalues, eigenvectors and eigenspaces of matrices and linear transformations
8. Prove various theorems and properties related to matrices, determinants, vector spaces, inner
product spaces, and linear transformations
Lesson Learning Outcomes (LLOs)
1. Define "set" and the related terminology.
2. Define "function" and the related terminology.
3. Explain what is meant by and be able to form the composition of functions.
4. Explain what is meant by a system of equations, a solution of the system, a consistent system, an
inconsistent system and a homogeneous system of equations.
5. Define "matrix."
6. Explain what is meant by an "m x n" matrix, a "square matrix of order n," the "(i,j) entry" of a
matrix and the "main diagonal" of a square matrix.
7. Define and determine the equality of two matrices, the sum of two matrices, the difference of two
matrices, the product of two matrices, and the product of the scaler and a matrix.
8. Use summation notation in the definition of matrix multiplication and proof of certain matrix
properties.
9. Define "transpose of a matrix" and find the transpose of a given matrix.
10. Make a formal or informal proof of various theorems concerning the above objects and
operations.
11. State all of the algebraic properties of matrix operations as discussed in class.
12. Prove selected algebraic properties of matrix operations as well as various theorems which are off
shoots of these properties.
13. State what is meant by the zero-matrix, by a diagonal matrix, a scalar matrix, and the identity
matrix of order n.
14. Define "upper triangular form" and "lower triangular form" for a matrix.
15. Define "singular" matrix, "nonsingular" matrix, and "inverse" of a matrix and find the inverse of a
given matrix when it arrives.
16. Prove various theorems concerning the objects mentioned in Objectives 13 - 15.
17. Explain the connection between singular and nonsingular matrices to the solution of a system of
equations.
18. Define "n by n" elementary matrices of type I, II, or III.
19. Prove selected theorems concerning the operation of elementary matrices on a given matrix.
20. Use elementary matrices to develop a technique for finding the inverse of a given matrix.
21. Explain what is meant by row-reduced echelon form for a matrix and transform a given matrix
into row-reduced echelon form.
22. Define the three elementary row operations on a matrix.
23. Explain what is meant by one matrix being row equivalent to a second matrix.
24. Prove various theorems concerning row equivalence and row-reduced echelon form.
25. Use matrix techniques discussed in class to solve systems of linear equations.
26. Define "real vector space" and explain the significance of each of the components of the
definition.
27. Give examples of a vector space.
28. Define "subspace of a vector space" and give examples.
29. Determine whether or not a given object is a vector space or subspace.
30. Use appropriate notation, work problems, and prove selected theorems involving vector spaces
and subspaces.
31. Define "linear combination" of a set of vectors.
32. State what is meant by a set of vectors "spanning" a vector space.
33. Explain what is meant by a linearly dependent or linearly independent set of vectors.
Math210 Syllabus
34. Define a "basis" for a vector space.
35. Explain what is meant by a nonzero vector space.
36. Define the dimension of a nonzero vector space.
37. Give examples, use appropriate notation, work problems, and prove selected theorems concerning
linear dependence and independence, bases, and dimensions of vector spaces.
38. Define “row space” and “column space” of an m by n matrix.
39. Explain what is meant by the row (column) rank of a matrix.
40. Discuss the structure of a linear system of equations.
41. Define the “determinant” of an n-by-n matrix and evaluate the determinant of a given matrix.
42. Discuss and prove the various properties of determinants and use these properties to aid in
solving problems involving determinants.
43. Define the "minor" of an element aij of a matrix A.
44. Define the "cofactor" of an element aij of a matrix A.
45. Explain and perform the process of finding a determinant by cofactor expansion.
46. Define the “adjoint” of a matrix A and find the adjoint of a given matrix.
47. Use appropriate notation and prove selected theorems which demonstrate the connection among
the inverse of a matrix, the determinant of a matrix, and the adjoint of a matrix.
48. Apply determinants in other selected situations as discussed in class.
49. Define the dot product of two vectors and discuss and/or prove its properties.
50. State the Cauchy-Schwarz inequality and the triangle inequality by cofactor expansion.
51. Define the “distance” between two vectors and what are “orthogonal” vectors.
52. Explain what is meant by an orthogonal set of vectors and an orthonormal set of vectors.
53. Define and calculate the scalar projection and vector projection of one vector on another.
54. Use appropriate notation, work problems, and prove selected theorems concerning inner products,
the Cauch-Schwarz and triangle inequalities, distance and orthogonality.
55. Discuss and use the Gram-Schmidt Probability for vectors.
56. Define "linear transformation" of a vector space V into a vector space W.
57. State what is meant by the “null space” and “range” of a linear transformation.
58. Explain what is meant by the matrix representation of a linear transformation.
59. Find the matrix representation of a given linear transformation.
60. Define the "sum," "scaler multiple" and "composition" of linear transformations and thereby
define a vector space of linear transformations.
61. State what is meant by a vector space of matrices.
62. Explain the concept of a coordinate vector with respect to an ordered basis.
63. Find how coordinate vectors transform under a change of basis.
64. Define "similar matrices.”
65. Give examples, use appropriate notation, work problems, and prove selected theorems concerning
rank of a matrix, linear transformations, null spaces, ranges, vector spaces of linear
transformations, and vector spaces of matrices.
66. Define "diagonalizable linear transformation" and give example space of matrices.
67. Define "eigenvalue" and "eigenvector" of a linear transformation, give examples, and find the
eigenvalues of eigenvectors of a given matrix.
68. State what is meant by the characteristic polynomial of a matrix.
69. Work problems based on the definitions mentioned in objectives 64-68 and theorems based on
those definitions.
70. Explain what is meant by a symmetric matrix and skew symmetric matrix, and determine whether
or not a given matrix is symmetric or skew symmetric.
71. Discuss and/or prove the theorems connecting diagonalization and symmetric matrices.
72. Define "Real Quadratic Form" and "equivalence" of real quadratic forms.
73. Explain what is meant by congruent matrices.
74. Use appropriate notation, work problems and prove selected theorems involving quadratic forms.
Math210 Syllabus
Final Course Grading Scale
Grade Percentage
A 90-100%
B 80-89%
C 70-79%
D 60-69%
F lower than 60%
Faculty Commitment
Faculty members are committed to providing a quality learning experience through thoughtful
planning, implementation, and assessment of course activities. They are also committed to being
readily available to students throughout the semester by returning e-mails and phone calls within
48 hours and to returning graded course work within a week. Furthermore, they are committed to
selecting appropriate course materials and making them available in an organized and timely
manner.
Student Commitment
For every credit hour a student is enrolled in, they should expect to spend at least 2 hours outside
of class studying, working on assignments, and preparing for class each week of the fifteen-week
semester. For example, for this three credit-hour class, students can expect to spend three hours
per week in class actively engaged in learning the material by participating in face-to-face
classes or viewing lectures and instructional material online. In addition, students should expect
to spend another six hours per week outside of class completing homework and assignments,
posting to discussion boards online, or studying for quizzes and tests. This means students should
spend a minimum of 9 hours per week engaged in achieving the learning outcomes for this
course. If you are not achieving your desired results in this class, you should consider increasing
your prep time outside of class, in addition to using available resources such as instructor office
hours and tutoring services.
By registering for this course, you commit yourself to active participation in course activities as
well as the submission of all assignments and exams on time. Furthermore, you commit to
accessing the course site and checking your JJC e-mail several times a week.
