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# Applied Linear Algebra

Course Code: MATH 2210
Faculty: Science & Technology
Department: Mathematics
Credits: 3.0
Semester: 15 Weeks
Learning Format: Lecture, Seminar, Tutorial
course overview

MATH 2210 is an introductory course in linear algebra with an emphasis on application to problems in engineering and science. Topics include vectors and geometry, systems of linear equations, matrices, subspaces, determinants, linear transformations, complex numbers, eigenvalues and eigenvectors, and orthogonality. Students with credit for MATH 2232 may not take this course for further credit.

### Course Content

•Vectors and Geometry: The geometry and algebra of vectors, dot product, lines and planes
•Systems of Linear Equations: Solution by row reduction, geometry of linear systems
•Subspaces of Rn: Subspaces, span, linear independence, basis, dimension
•Matrices: Matrix operations, algebra, inverse; special forms, rank, fundamental subspaces of a matrix
•Linear Transformations: Matrices as transformations, geometry of linear transformations, kernel and range, composition, invertibility, change of basis
•Determinants: Calculating determinants, properties of determinants
•Complex Numbers: The complex plane, arithmetic, polar form, De Moivre’s formula, Euler’s formula
•Eigenvalues and Eigenvectors: Properties and geometry, calculating complex eigenvalues and complex eigenvectors, similarity and diagonalization
•Orthogonality: Orthogonal and orthonormal basis, projections, orthogonal subspaces and complements, least-squares approximation

### Methods of Instruction

Lectures and tutorials

### Means of Assessment

Evaluation will be carried out in accordance with Douglas College policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on the following:

• Quizzes: 0-20%
• Tests: 20-70%
• Assignments: 0-15%
• Tutorials: 0-10%
• Final Examination: 30-40%

### Learning Outcomes

Upon completion of MATH 2210, the successful student will be able to:

• Perform basic arithmetic computations with vectors in Euclidean two-space, R2, and three-space, R3 and generalize these computations to Euclidean n-space, Rn
• Define geometric aspects of vectors in R2 and R3 such as length, the dot product, the angle and distance between two vectors, and orthogonality, and generalize these definitions to Rn
• Solve problems using the various forms of the equations of lines and planes in R3
• Express a system of linear equations in vector form or matrix form, and convert between each form
• Solve systems of linear equations using Gaussian elimination and row reduction
• Describe properties of solutions to homogeneous systems of linear equations, and the connection between solutions of homogeneous and non-homogeneous linear systems
• Give a geometric interpretation of subspaces of R2 and R3 and generalize these interpretations to Rn
• Define span and linear independence for a set of vectors, and determine if a set of vectors in Rn is linearly independent
• Use the concepts of subspaces, linear independence and span to give a geometric interpretation of solutions to systems of linear equations
• Define basis and dimension, and determine if a set of vectors is a basis for a subspace of Rn
• Perform basic operations on matrices: addition, scalar multiplication, matrix multiplication, transpose, powers, and apply appropriate properties of matrix algebra when performing these operations
• Define a matrix inverse and apply properties of matrix inverses
• Determine the inverse of a matrix by Gaussian elimination
• Define and determine the rank of a matrix
• Determine the basis and dimension for the fundamental subspaces of a given matrix
• Define a linear transformation and determine if a given transformation is a linear transformation
• Interpret linear transformations in terms of matrices and determine the standard matrix for a linear transformation from Rn to Rm
• Determine the matrices that describe a rotation, a shear, a dilation or contraction, and a reflection in R2; Given a 2 x 2 matrix, describe the transformation in terms of the foregoing matrices
• Determine if a linear transformation is one-to-one, onto or invertible
• Determine the kernel, range, rank and nullity of a linear transformation and interpret these in terms of the fundamental subspaces of a matrix
• Form composite transformations from given linear transformations
• Determine the coordinates of a vector in a given basis and determine the transition matrix for a change of basis
• Evaluate the determinant of an n X n matrix
• Apply basic properties of determinants when evaluating the determinant of a matrix
• Discuss the solvability of a system of linear equations using determinants
• Define a complex number, a complex conjugate and the complex plane
• Perform basic arithmetic computations with complex numbers
• Express complex numbers in polar form, and work with DeMoivre’s formula and Euler’s formula
• Define and give a geometric interpretation of eigenvalues and eigenvectors of a matrix
• Determine the characteristic polynomial, eigenvalues and corresponding eigenspaces of a given matrix, including those involving complex numbers
• Define similar matrices and use this property to diagonalize a square matrix
• Define standard basis, orthogonal basis, orthonormal basis, orthogonal projections, orthogonal matrix, orthogonal complement
• Calculate the projection of one vector onto another in Rn
• Convert a basis into an orthonormal basis using the Gram-Schmidt Process (max of three vectors) in Rn
• Orthogonally diagonalize a symmetric matrix
• Use orthogonal projection to find the least-squares solution of a system of linear equations
• Solve application problems related, but not limited, to: electrical network analysis, vector differential equations, dynamical system computer graphics, network analysis traffic flow

course prerequisites

MATH 1120 or MATH 1130

### Equivalencies

Courses listed here are equivalent to this course and cannot be taken for further credit:

• Students with credit for MATH 2232 may not take this course for further credit.

curriculum guidelines

Course Guidelines for previous years are viewable by selecting the version desired. If you took this course and do not see a listing for the starting semester/year of the course, consider the previous version as the applicable version.

course schedule and availability
course transferability

Below shows how this course and its credits transfer within the BC transfer system.

A course is considered university-transferable (UT) if it transfers to at least one of the five research universities in British Columbia: University of British Columbia; University of British Columbia-Okanagan; Simon Fraser University; University of Victoria; and the University of Northern British Columbia.

For more information on transfer visit the BC Transfer Guide and BCCAT websites.

assessments