# Linear Algebra

Prerequisites: Mathematics 1D with a grade of C or better.

Advisory: English Writing 211 and Reading 211 (or Language Arts 211), or English as a Second
Language 272 and 273.

Five hours lecture.

Linear algebra and selected topics of mathematical analysis .

II. Course Objectives

A. Solve and analyze systems of linear equations using matrices and matrix theory

B. Develop understanding and use of n-dimensional vectors and vector operations.

C. Define and discuss vector spaces and vector sub-spaces and find their bases and dimensions

D. Establish understanding of linear transformations and their geometry and find their matrix
representation.

E. Define eigenvalues and eigenvectors and use them to diagonalize square matrices and solve related
problems

F. Utilize methods of linear algebra to solve application problems selected from engineering, science
and related fields

G. Investigate the history and development of linear algebra as it relates to scientific, social and cultural
activities throughout the world.

III. Essential Student Materials

IV. Essential College Facilities

Computer lab (optional)

V. Expanded Description : Content and Form

A. Solve and analyze systems of linear equations using matrices and matrix theory

1. Convert systems of equations to matrix equations and produce augmented and coefficient
matrices.

2. Use row operations to put matrices into row echelon and row reduced echelon forms

3. Apply the row echelon form of a matrix to classify a system of linear equations as
consistent/inconsistent, dependent/independent.

4. Use row reduced form of augmented matrices to write solutions in point and parametric forms.

5. Investigate and solve application problems from, but not limited to, engineering, science and
related fields.

B. Develop understanding and use of n-dimensional vectors and vector operations.

1. Explore n-dimensional vectors and basic vector operations with emphasis on 2D and 3D vectors.

2. Use vector inner product to determine angles between two vectors and orthogonality.

3. Apply the algebra of 2D and 3D vectors to study lines and planes in 3D space.

C. Define and discuss vector spaces and vector sub-spaces and find their bases and dimensions

1. Develop an understanding of Euclidian n-dimensional space, norm, Cauchy-Schwartz and
triangle inequalities

2. Investigate general linear spaces

3. Explore linear dependence and independence

5. Determine basis and dimension of vector spaces.

6. Change basis and investigate change of base matrices.

7. Use the Gram-Schmidt algorithm to orthonormalize a set of vectors.

8. Apply the Gram-Schmidt algorithm to investigate special polynomials (like Legendre) (optional)

D. Establish understanding of linear transformations and their geometry and find their matrix
representation.

1. Analyze the geometric implications of linear transformations in 2-and 3-space.

2. Study properties of linear transformations

3. Identify the fundamental subspaces of linear transformations.

4. Investigate the nullity and rank of linear transformations.

5. Construct bases of the image of a linear transformation described by a matrix.

6. Investigate the composition and inverse of linear transformations

7. Construct a matrices of general linear transformations using non-standard bases.

E. Define eigenvalues and eigenvectors and use them to diagonalize square matrices and solve related
problems

1. Define eigenvalues and eigenvectors of a matrix

2. Use the characteristic equation to find eigenvalues of a matrix

3. Find the eigenspace of a matrix

4. Investigate the conditions for diagonalization and orthogonal diagonalization of a matrix

5. Use standard procedures to diagonalize and orthogonally diagonalize matrices

6. Choose application problems from areas such as, but not limited to, dynamical systems, Markov
chains, cryptography, and game theory.

F. Utilize methods of linear algebra to solve application problems selected from engineering, science
and related fields

1. Iterative methods for solving linear systems such as Gauss-Seidel method.

2. The power method for finding eigenvalues of a matrix and its application to internet search
engines.

3. Use of projection matrices for the general least squares approximations.

4. Transform equations of general quadric surfaces into standard form

G. Investigate the history and development of linear algebra as it relates to scientific, social and cultural
activities throughout the world.

1. Investigate the history and development of linear algebra from times of ancient China to present
day.

2. Examine the history of matrix theory and major individual contributions

3. Explore the history and applications of linear algebra among various human societies.

VI. Assignments

B. Problem-solving activities assigned on daily or weekly basis

C. Projects that may include the use of technology. (optional)

VII. Methods of Instruction

Methods of instructions may include but not limited to
Lecture and visual aids
Discussion and problem solving as a class activity
Collaborative learning and small group exercises
Collaborative projects
Use of various technologies including graphing utilities and computer labs.

VIII. Methods of Evaluating Objectives

A. Homework/quizzes

B. A minimum of three one hour written examinations or two one hour written examinations and an
individual or group project

C. Two-hour comprehensive final exam

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