Finite dimensional vector spaces, linear independence, basis, systems of linear equations, linear transformations, matrices, LU factorization, change of basis, similarity of matrices, eigenvalues and eigenvectors, applications, quadratic forms, symmetric and orthogonal matrices, canonical forms, and introduction to infinite dimensional vector spaces.
64.000-72.000 Total Hours
Total Hours
128.000-144.000 Total Hours
64.000-72.000 Total Hours
Prerequisite: MATH 160 Prerequisite or Corequisite: None Concurrent Corequisite: None Course Advisories: None Limitation on Enrollment: None
Course Objectives:
Compute the matrix for a linear transformation.
Solve Ax=b by elimination, the invertibility of A, or LU factorization.
Determine basis and dimension of vector spaces, especially for column and null spaces of A and the transpose of A.
Project a vector onto a subspace.
Use the Gram-Schmidt procedure to find an orthonormal basis.
Compute determinants using their properties.
Find eigenvalues and eigenvectors of a matrix.
Diagonalize a matrix, compute its powers, and its exponential matrix.
Write quadratic forms as matrix products using real, symmetric matrices A.
Apply Linear Algebra to problems such as systems of differential equations, finite difference equations, graphs and networks, Markov matrices, Fourier matrix, Fast Fourier Transform, and linear programming.
Student Learning Outcomes
MATH210 SLO1 - Solve Ax=b using a variety of methods such as Gaussian elimination and inverting the matrix A.
MATH210 SLO2 - Identify a linear transformation and find the eigenvalues and corresponding eigenspaces.
MATH210 SLO3 - Diagonalize a matrix or determine why it is not possible to do so.
MATH210 SLO4 - Orthogonalize bases using the Gram-Schmidt process and produce unique representations in terms of these bases.
MATH210 SLO5 - Prove Linear Algebra theorems and corollaries using appropriate writing techniques involving topics such as linear independence of vectors; properties of subspaces; linearity, injectivity, and surjectivity of functions; and properties of eigenvalues and eigenvectors.
MATH210 SLO6 - Apply Linear Algebra to problems in the sciences.
Systems of Linear Equations and Matrices
Gaussian Elimination and LU factorization
Homogeneous Systems of Linear Equations
Matrices including special matrices such as Diagonal, Triangular, Elementary, and Symmetric
Matrix Operations including the transpose of a matrix
Rules of Matrix Arithmetic
Methods for finding the Inverse of a Matrix
Techniques For Solving Linear Systems such as Gaussian and Gauss-Jordan Elimination and Inverse Matrices
Relationship between Coefficient Matrix, Invertibility, Solutions to a System of Linear Equations, and the Inverse Matrix
Determinants
Evaluating Determinants by Row Reduction
Properties of the Determinant Function
Cofactor Expansion; Cramer's Rule
Vectors in 2-Space and 3-Space
Introduction to Vectors (Geometric)
Norm of a Vector, Angle between vectors, Orthogonal vectors
Vector Arithmetic
Inner Product; Projections
Cross Product
Lines and Planes in 3-Space
Vector Spaces
Euclidean n-Space
General Vector Spaces
Subspaces
Linear Independence and Dependence
Basis and Dimension
Finding basis for Row, Column, Null, and Left Null Spaces; Rank; Nullity
Inner Product Spaces
Length, Orthogonality, and Angle in Inner Product Spaces
Orthogonal and Orthonormal Basis; Gram-Schmidt Process; QR factorization
Coordinates; Change of Basis
Linear Transformations
Definition of Linear Transformation
Properties of Linear Transformations; Kernel and Range
Linear Transformations and Inverse Linear Transformations
Matrices of General Linear Transformations
Similarity
Eigenvalues, Eigenvectors, Eigenspaces
Definitions and Introduction
Diagonalization
Orthogonal Diagonalization of Symmetric Matrices
Applications
Approximation Problems; Fourier Series
Quadratic Forms
Other applications as time and interest allow
Methods of Instruction
Lecture
Lecture is the primary activity, along with student problem solving. Students are expected to work outside of class on reading the text and on assigned exercises that may include the use of a computer algebra system, and supplemental reading as determined by the instructor.
-Problem: Suppose k and m are two distinct eigenvalues of a 2 by 2 matrix A. Prove that the columns of A - kI must be multiples of an eigenvector for m.
1. Appropriate Readings: Students are required to read assigned chapters in texts.
2. Writing Assignments: Students must work assigned mathematical problems requiring the understanding of abstract ideas.
3. Appropriate Outside Assignments: Students will be expected to spend a sufficient amount of time outside of class to practice techniques taught during class time, read assigned materials, and complete frequent homework and computer assignments.
4. Appropriate Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve analyzing information, recognizing concepts in new contexts, and drawing analogies. They must also analyze logical arguments for validity and write proofs of their own using both inductive and deductive reasoning within a logical system. Students will also learn to use software tools in solving problems from linear algebra.
A student's grade will be based on multiple measures of performance in the solving of problems, designing of mathematical models, preparation and analysis of graphs, and analysis of logical arguments. Such measures will include at least four one-hour exams and a comprehensive final examination requiring demonstrations of problem solving skills. In addition, instructors may make use of quizzes, written homework assignments, or other appropriate means to judge a student's dexterity with mathematical skills and familiarity with mathematical vocabulary and methods of proof.
Linear Algebra and Its ApplicationsLay, D. C., Addison-Wesley, 2015Introduction to Linear AlgebraStrang, G., Wellesley - Cambridge Press, 2016