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SMC|Academic Programs|Mathematics|Math 13 - Linear Algebra

Math 13 - Linear Algebra

Course Details

Description

Topics include matrices and linear transformations, abstract vector spaces and subspaces, linear independence and bases, determinants, systems of linear equations, and eigenvalues and eigenvectors.

Prerequisite

Math 8

How It Transfers

UC, CSU IGETC AREA 2 (Mathematical Concepts)

Textbook

Grossman, Elementary Linear Algebra, 5th Edition, Cengage, 1994  (Mazorow)

Larson, Edwards & Falvo, Elementary Linear Algebra, 6th Edition, Houghton Mifflin, 2009  (Nestler)

Mathematics Skills Associated With This Course

Entry Level Skills

Skills the instructor assumes you know prior to enrollment in this course

  • Solve systems of linear equations using Gaussian Elimination. 
  • Write the equation of a line in parametric form. 
  • Prove mathematical statements by methods including proof by contradiction and mathematical induction.
  • Integrate and differentiate functions including functions defined by infinite series. 
  • Evaluate, manipulate, and interpret summation notation. 
  • Given a function, over an interval, be able to prove algebraically the existence of its inverse function by formally proving the function is one to one. Be eligible for English 1.
Course Objectives

Skills to be learned during this course

  • Apply the concepts and theorems of linear algebra to show the consequences of a given definition. 
  • Perform matrix computations and apply matrix algebra. 
  • Express a matrix as a product of elementary matrices and an upper triangular matrix.
  • Compute the inverse, if possible, of a square matrix, and express it as a product of elementary matrices. 
  • Solve any size system of linear equations using Gaussian Elimination, and, where necessary, express solutions using parameters or as a linear combination of basis vectors. 
  • Apply fundamental determinant theorems.
  • Prove whether or not a set and operations form a vector space (or subspace). 
  • Apply the concepts of linear independence and spanning to find a basis for a vector space. 
  • Prove whether or not a function between two vector spaces is a linear transformation or isomorphism. 
  • Find the matrix representation of a linear transformation with respect to two given ordered bases. 
  • Express the kernel and range of a linear transformation as a span of basis vectors. 
  • Compute the eigenvalues for a matrix, find a basis for the corresponding eigenspaces, and where possible, diagonalize the matrix. 
  • Use the Gram-Schmidt process to compute an orthonormal basis of a space.