Liberal Arts Division 
Catalog Description: An elementary introduction to linear algebra topics including linear equations and matrices, determinants,
independence and basis, vector spaces and subspaces, four fundamental subspaces, orthogonality, linear transformations and
eigenvalues and eigenvectors. Applications of
linear algebra are included.
Prerequisites: MAT151 with minimum C grade
Required
Materials:
|
Text: Linear Algebra with Applications, 7th Edition Nicholson, 2013 ISBN is 9781308393834 for custom book |
A graphing calculator is
required. TI – 83/84/86 is
strongly recommended |
Instructor
Contact Info:
|
E-mail: porterr@mccc.edu |
Office: LA
129 |
Office Hours (see webpage) |
|
Web Page:
http://www.mccc.edu/~porterr |
Phone:616-2841 |
Grading: “No make-up exams!
The Final will used to calculate grades for valid excuses.”
**Must Receive a 50% on Final to pass course!** |
93%-100%........A 90%-92%..........A- 87%-89%..........B+
A curve may be applied 83%-86%..........B
to low exam grades, but 80%-83%..........B-
not to high exams. 77%-79%..........C+ 70%-76%..........C 60%-69.5%.......D <59.5%.............F |
Topics:
|
|
|
|
|
|
1 |
1/20 |
|
Vectors,
Lengths, Dot Products |
|
2 |
1/22 |
|
Vectors,
Linear Equations, Elimination |
|
3 |
1/27 |
|
Rules
for Matrices |
|
4 |
1/29 |
Inverse Matrices, Factorization |
|
|
5 |
2/3 |
Transposes, Permutations |
|
|
6 |
2/5 |
Properties of Determinants |
|
|
7 |
2/10 |
|
|
|
8 |
2/12 |
|
Spaces
of Vectors |
|
9 |
2/17 |
|
Orthogonality, Nullspace of A |
|
10 |
2/19 |
|
Rank,
Row Reduced Form |
|
11 |
2/24 |
|
Independence,
Basis. Dimension |
|
12 |
2/26 |
|
Orthogonality of Subspaces |
|
13 |
3/3 |
|
Projections,
Least Squares Approx. |
|
14 |
3/5 |
|
Orthogonal
Basis, Gram-Schmidt |
|
15 |
3/10 |
|
|
|
16 |
3/12 |
|
|
|
17 |
3/24 |
|
Cramer’s
Rule, Inverses, Volumes |
|
18 |
3/26 |
|
Intro
to Eigenvalues |
|
19 |
3/31
|
|
Diagonalizing a Matrix |
|
20 |
4/2 |
|
Linear
Transformations |
|
21 |
4/7 |
|
Matrix
of a Linear Transformation |
|
22 |
4/9 |
|
Change
of Basis |
|
23 |
4/14 |
|
|
|
24 |
4/16 |
|
Singular
Value Decomposition |
|
25 |
4/21 |
|
More
SVD |
|
26 |
4/23 |
|
Diagonalization, Pseudoinverse |
|
27 |
4/28 |
|
More
7.4 |
|
28 |
4/30 |
|
|
|
29 |
5/5 |
|
|
|
30 |
5/7 |
|
FINAL EXAM |
| 5/12 | FINAL EXAM 6-9PM in LA210 |
Please
show ALL your own work for full credit- even when using a calculator.
At the end of Unit 1, the student should be able to:
calculate the dot product between vectors. (Course Goal 1)
calculate the length of a vector. (Course Goal 1, Gen Ed Goal 2)
explain the concept of orthogonality of vectors. (Course Goal 1)
create a unit vector from a given vector. (Course Goal 1)
demonstrate that one vector is a linear combination of given vectors. (Course Goal 1)
apply matrix operations. (Course Goal 1)
set up matrix operations using proper technology. (Course Goal 1,12)
2. DETERMINANTS AND INVERSES (2 weeks) Chpt 2 & 3
At the end of Unit 3, the student should be able to:
calculate the inverse of a matrix. (Course Goal 2, 12)
solve systems by using row reduction and LU factorization. (Course Goal 2, 12)
recognize the connection between the elimination process and factoring a matrix.
(Course Goal 2)
define determinant. (Course Goal 10)
calculate the determinant of a square matrix and interpret it in terms of invertibility of a matrix.
(Course Goal 4)
apply the basic properties of determinants. (Course Goal 4, 12)
apply Cramer’s Rule to solve systems of equations and volume problems. (Course Goal 4, 12)
3. VECTOR SPACES AND SUBSPACES (3.5 weeks) = chpt 5 & chpt 6
At the end of Unit 3, the student should be able to:
explain the defining properties of a vector space. (Course Goal 3)
construct examples of vector spaces. (Course Goal 3)
explain why a set defined with the necessary operations is or is not a vector space.
(Course Goal 3)
explain why a subset of a given vector space is or is not a subspace. (Course Goal 3)
define the span of a set of vectors. (Course Goal 3)
determine if a collection of vectors from a given vector space is a spanning set for the vector space.
(Course Goal 3, 12)
define linear independence. (Course Goal 3 and 5)
calculate whether or not a given set of vectors is linearly independent. (Course Goal 5,12)
calculate the rank of a given matrix. (Course Goal 5, 12)
explain the defining properties of a basis for a vector space. (Course Goal 5)
determine if a given set of vectors from a vector space is or is not a basis for the space.
(Course Goals 5, 12)
explain what is meant by the dimension of a vector space. (Course Goal 5)MAT208 Course Outline Fall 2012
4
state the properties of subspaces and the relationships among the four fundamental subspaces of a
matrix. (Course Goals 3 and 5, Gen Ed Goals 1 and 2)
explain why the equation Ax=b is consistent if and only if b is in the column space of A. (Course
Goals 3 and 5, Gen Ed Goals 1 and 2)
discuss how linear independence, spanning sets, basis and dimension are related. (Course Goals 5
and 12)
4. ORTHOGONALITY AND LEAST SQUARES (2 weeks) = Chpater 5 & 8
At the end of Unit 4, the student should be able to:
state the properties of orthogonal matrices. (Course Goal 7)
derive the normal equations for a least squares problem and solve it. (Course Goal 7)
explain what condition must be satisfied for the normal equations to have a unique solution.
(Course Goal 6)
apply least squares approximations to problems to minimize errors. (Course Goals 6 and 12)
calculate the error in a least squares problem. (Course Goals 6 and 12)
apply the Gram-Schmidt process to construct an orthonormal set of vectors. (Course Goal 7, 12)
construct the QR factorization of a matrix. (Course Goal 7 and 12)
apply the QR factorization to least-squares problem. (Course Goals 6,7, and 12)
compare the methods of matrix factorization studied so far. (Course Goals 2 and 7)
5. EIGENVALUES AND EIGENVECTORS (3 weeks) = chapter 8
At the end of Unit 5, the student should be able to:
define eigenvalue and eigenvector. (Course Goal 10)
calculate the characteristic equation of a square matrix and solve to find eigenvalues.
(Course Goal 10, 12)
calculate the associated eigenvectors of a square matrix. (Course Goal 10, 12)
explain how the eigenvalues of similar matrices are related. (Course Goals 4 and 10)
analyze the relationship among determinants and number of eigenvalues, determinants and the
product of eigenvalues, and the trace and the sum of eigenvalues. (Course Goals 4, 10, 12)
explain the application of diagonalizing a matrix to the matrix exponential. (Course Goal 9)
define pseudoinverse for a non-square matrix. (Course Goal 11)
construct the singular value decomposition of an mxn matrix. (Course Goal 9, 12)
explain the relationships among the columns of U and V of
T A U V
, the singular value
decomposition of an mxn matrix A, and the four fundamental subspaces associated with A.
(Course Goals 5, 9, 10, 12)
summarize the use of the singular value decomposition in applications such as Web search engines
and image processing. (Course Goal 9, 12)
apply the singular value decomposition and pseudoinverse to solve least squares problems. (Course
Goals 9,11, and 12)
conclude, through the use of technology, that the singular value decomposition is necessary for
factoring non-square matrices. (Course Goals 9 and 12)MAT208 Course Outline Fall 2012
6. GENERALIZED VECTOR SPACES AND LINEAR TRANSFORMATIONS (2.5 weeks) = chpt 7 & chpt 9
At the end of Unit 6, the student should be able to:
state the definition of a linear transformation from a vector space V to another vector space W.
(Course Goal 8)
give examples of linear transformations. (Course Goal 8)
calculate the kernel and range of a linear transformation (Course Goal 8)
identify a linear transformation and find and use its matrix representation. (Course Goal 8,12)
illustrate the process of change-of-basis by building on previous work and definitions. (Course
Goal 8)
calculate the matrix representation of a linear transformation from a vector space V to a vector
space W with respect to two given bases. (Course Goal 8)
examine the geometry of linear transformations. (Course Goal 8)