Unit 1
Review

Introduction

This unit contains a quick review of Mathematics 270: Linear Algebra 1. If you have recently completed this course, or an equivalent university course in linear algebra, you may wish to skip immediately to the Tutor-marked Quiz at the end of this unit, and then proceed to Unit 2. You should, however, at least read through this unit and work some of the exercises to refresh your memory. We advise that students who have not taken a university-level linear algebra course for some time go through this unit carefully, working all of the problems, as a review of both concepts and computational skills.

Objectives

When you have completed this unit, you should have a clear understanding of the material covered in Mathematics 270: Linear Algebra I, and should be able to

  1. recognize and use the index and summation notation for writing systems of linear equations, denoting matrices and vectors, and indicating vector spaces and subspaces.
  2. perform calculations involving matrix addition, subtraction, and multiplication; and compute inverses using elementary matrices.
  3. define the augmented matrix for a system of linear equations, and solve this system using Gauss-Jordan elimination.
  4. determine whether or not a system of linear equations is consistent.
  5. determine whether or not a matrix is invertible, and compute the inverse of an invertible matrix.
  6. compute the determinant of a matrix, and state from memory the important theorems about determinants.
  7. compute the adjoint matrix of a given matrix, and express the inverse of a matrix in terms of its adjoint.
  8. solve a system of linear equations using Cramer’s Rule.
  9. carry out basic vector operations in two and three dimensions, determine the equations of straight lines parallel to given vectors, and determine equations of planes in three dimensions.
  10. generalize what you know of two- and three-dimensional Euclidean space to n-dimensional Euclidean space.
  11. define a linear transformation, and compute the standard matrix for any given linear transformation.
  12. define and compute the eigenvalues and eigenvectors of a square matrix.
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