MATH 3435– Content Standards

Title: Introductory Linear Algebra

Catalog course description: Prerequisites: Math 2215 and Math 3000.
Theory and applications of matrix algebra and linear transformations. Topics include linear equations, vector spaces, matrices, subspaces, and bases.  

Goals

To provide students with a solid background in linear algebra and basic matrix theory, including applications.  

Course Content Standards  

The following standards are offered as guidelines for assessing student progress, judging the effectiveness of instructional programs, and developing curricular units. The subject matter outlined in these standards represents the minimum knowledge in which a student should demonstrate proficiency at the successful completion of the course.  

  • Students will be able to identify a system of linear equations and form the augmented matrix for the system.  
  • Students will be able to identify when a matrix is in row echelon form or reduced row echelon form.  
  • Students will be able to identify when an augmented matrix in row echelon form corresponds to an inconsistent system, a system with a single solution, or a system with multiple solutions.  
  • Students will be able to use elementary row operations to reduce an augmented matrix to row echelon form and to use the form, together with back-substitution, to solve the corresponding system.  
  • Students will be able to perform algebraic operations on vectors in n-dimensional space.  
  • Students will be able to interpret the geometric properties of vectors in R n and of algebraic operations on vectors in R n.  
  • Students will know the definitions of a linear combination and of the span of a set of vectors and the geometric significance of a vector being in the span of a set of vectors.  
  • Students will be able to represent a set of linear equations as a combination of the columns of the system matrix A and also as the matrix-vector product Ax.  
  • Students will recognize consistent systems as those in which the right hand side is a combination of the columns of the system matrix A.  
  • Students will be able to computationally determine if a given set of vectors is linearly independent and determine if a given vector is in the span of a set of vectors.  
  • Students will know the definition of a linear transformation and will be able to represent linear transformations as matrices.  
  • Students will be able to identify one to one and onto linear transformations.
  • Students will be able to apply the theory of linear systems to simple applied problems.  
  • Students will be able to apply basic matrix operations, including products, sums and transposes.  
  • Students will be able to determine if two matrices are inverses of each other.  
  • The student will be able to deduce the uniqueness of solutions from invertibility.  
  • Students will be able to compute the inverse matrix using elementary row operations.  
  • Students will be able to apply the equivalent conditions of the invertible matrix theorem to determine if matrices are invertible.  
  • Students will be able to apply the definition of the determinant to compute the determinant of a matrix.  
  • The student will know the effect of elementary row operations on the determinant.  
  • Students will be able to compute determinants using elimination.  
  • Students will know the properties and definition of a vector space and be able to apply these properties in computations involving vectors.  
  • Students will be able to tell if a given set is a subspace.  
  • Students will be able to find the null space and column space of a matrix and be able to relate them to kernel and range of a linear transformation.  
  • Students will be able to apply the definition of linear independence and to recognize linearly independent and linearly dependent sets of vectors.  
  • Students will be able to recognize a basis for a subspace and be able to construct a basis for the span of a set of vectors.  
  • Students will be able to define a coordinate system with a basis and be able to find the coordinates of a vector with respect to a given basis.  
  • Students will be able to change a basis and represent the basis change as a matrix.  
  • Students will be able to determine the dimension of a subspace.  
  • Students will be able to compute the rank of a matrix and to relate the rank to the dimension of the null and column spaces of a matrix.  
  • Students will be able to describe a Markov chain using its probability transition matrix.  
  • Students will be able to find the steady state of a Markov chain.