Weekly Distribution:
- Lecture/Seminar: 4 hours/week
- Tutorial 1 hour/week
Lectures and tutorials
•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
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
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%
Consult the Douglas College Bookstore for the latest required textbooks and materials. Example textbooks and materials may include:
- Linear Algebra, A Modern Introduction, David Poole, current edition, Brooks/Cole/Cengage
- Contemporary Linear Algebra, Howard Anton, Robert C. Busby, current edition, Wiley
- Elementary Linear Algebra, Larson and Falvo, current edition, Brooks/Cole/Cengage
- A First Course in Linear Algebra, Ken Kuttler, An Open Text by Lyryx
- Linear Algebra with Applications, W. Keith Nicholson, An Open Text by Lyryx
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.