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Master Course Outline
MATH 220
Linear Algebra

Credits: 5
Clock Hours per Quarter: 50

AA Discipline: [Quantitative] [Natural Sciences]

Lecture Hours:50

Designed for students planning studies in mathematics, engineering, computer science, and physics. Topics include systems of linear equations, matrices, determinants, eigenvalues, eigenvectors, vector spaces, linear transformations, orthogonality, and diagonalization. Prerequisite: Grade C- or higher in MATH& 153 or permission of the Mathematics Department.

Intended Learning Outcomes
  • Systems of linear equations:
    a. Represent a linear system as an augmented matrix and use Gaussian elimination find the solution(s) to the system. Develop the notions of pivot columns, free variables, row echelon form, and reduce row echelon form of a matrix.
    b. For linear systems two and three unknowns develop and understand the geometry of a linear system and its solution(s).
    c. Use a computer algebra system (CAS) to solve linear system of any size using Gaussian elimination/row reduction.
  • Vectors
    a. Develop a visual understanding of vectors in R2 and R3.
    b. Develop the notion of a linear combination of vectors algebraically and geometrically along with the span of a set of vectors.
    c. Develop the notions of the vector form of a line, the vector form of a plane, and the decomposition of the solution to a linear system as a linear combination of vectors.
    d. Develop thoroughly the notion of linear independence of a set of vectors. Emphasize the meaning of linear independence geometrically for vectors in R2 and R3.
  • Matrices
    a. Develop the notion of a matrix times a vector as being a linear combination of the columns of the matrix using the vector entries as weights.
    b. Establish the connections between linear systems, vector equations, and matrix equations. Specifically the equivalent representation of a linear system as a vector equation or a matrix equation.
    c. Develop matrix multiplication by generalizing the matrix-vector product to the matrix-matrix product.
    d. Develop fully the rules of matrix algebra.
    e. Develop fully the notion of multiplicative inverses for matrices. When does matrix have an inverse and how is the inverse of a matrix determined.
    f. Develop competence in using the CAS to work with matrices, vectors, etc.
    g. Introduce the "Invertibility Theorem" establishing equivalences amongst matrices, vectors, and solutions to linear systems.
  • Linear Transformations
    a. Define a linear transformation from RM to Rn. Establish that every linear transformation is represented by a matrix transformation. Develop the linear properties and algebraic operations on linear transformations. Define the kernel and range of a linear transformation.
    b. Investigate fully the geometric linear transformations in R2 and R3; dialations/contractions, shears, rotations, reflections, rotations, and projections.
    c. Develop competence in using a CAS to work with linear transformations.
  • Vector Spaces
    a. Develop the notions of vector spaces, subspaces, basis, dimensions for Rn vector spaces. Emphasize geometry for R2 and R3. Develop the notion of coordinates relative to a basis and how to change between difference basis for a vector space.
    b. Develop a subspaces associated with a matrix - row space, column space, and null space. Include techniques for finding basis for each of these spaces.
  • Determinants: Development of the basic properties of determinants and how to find the determinant of a matrix using a CAS.
  • Eigenvalues, eigenvectors, eigenspaces, and eigenbasis
    a. Develop the notions of the eigenvalues, eigenvectors, and eigenspaces of a matrix and emphasize the geometry for 2 by 2 and 3 by 3 matrices.
    b. Define and develop the properties of the characteristic polynomial of a matrix.
    c. Develop the notion of diagonalization of a matrix and similarity.
    d. Develop competence in using a CAS to find eigenvalues and eigenvectors for a matrix, to determine the characteristic polynomial of a matrix, and to diagonalize a matrix.
  • Orthogonality
    a. Develop the notion of orthogonality for a set of vectors and an orthogonal/orthonorman basis for Rn.
    b. Develop the nortion of orthogonal projections (into a subspace) and the Gram-Schmidt process for finding an orthogonal basis of a subspace.
    c. Develop the notion of orthogonal matrices, orthogonal complements, and othrogonal projections.
    d. Use orthogonailty to develop least-squares solutions to inconsistent linear systems. Include the normal equations to the linear system. Use least-squares solutions to solve a data fitting problem.
    e. Develop competence in using a CAS to work with orthogonality concepts, Gram-Schmidt, and finding least-squares solutions to inconsistent linear systems.

  • Syllabi Listing See ALL Quarters
    Year Quarter
    MATH 220
    Winter 2011
    Julianne Sachs
    MATH 220
    Winter 2009
    Eric Schulz
    MATH 220
    Winter 2008
    MATH 220
    Winter 2006
    Eric Schulz
    MATH 220
    Winter 2005
    Eric Schulz

    Two Year Projected Schedule

    Year One* Year Two**

    *If fall quarter starts on an odd year (2003, 2005, etc.), it's Year One.
    If fall quarter starts on an even year (2002, 2004, etc.), it's Year Two.