MA251-B Spring 2007 Final Exam Study Guide

This document is intended to offer some guidelines to help you prepare for the final exam in MA251.

It is not intended to be either a slightly modified version of the exam or an exclusive list of topics the exam will cover.

As noted in class, the exam will be cumulative but will not include any in-depth questions on the material in chapter 4. It is reasonable to expect that you will remember one or two major points from yesterdays lecture such as the fact that homeomorphism is another name for an invertible linear transformation.

Algebraic Rules

This includes:

• Facts A.2 and A.5 and Definitions A.1, A.3, A.4, A.6, A.7, and A.8 (Appendix A)
• Definition 1.3.5, 1.3.6, and 1.3.7 and Facts 1.3.8 1.3.9, and 1.3.10
• Definition 2.4.1 and Facts 2.4.2 through 2.4.11

Systems of Equations

• Determine whether a system of eqations is consistent or not
• Determine the number of solutions a consistent system has (one or infinitely many)
• Solve a system of equations using Gauss-Jordan reduction
• Solve a system of equations using Cramer's Rule (Fact 6.3.9)
• State the general solution of a system with many solutions in terms of free variables
• Know the relationship between the rank of the coefficient matrix A and the numbers of free and leading variables

Linear Transformations

• Be familiar with the definitions relating to linear transformations and basic facts about linear transformations:
  • Definition 2.1.1, Facts 2.1.2 and 2.1.3
• Be familiar with the special cases of linear transformations we studied, to the point of being able to recognize which one a given matrix corresponds to (section 2.2):
  • Scalings
  • Projections
  • Reflections
  • Rotations
  • Horizontal and Vertical Shears
• Be familiar with the terminology and concepts related to the inverse of a linear transformation, and computational algorithms for finding an inverse of a matrix:
  • Definitions 2.3.1, 2.3.3, Facts 2.3.4, 2.3.5, 2.3.6 and especially Summary 7.1.5

Subspaces

• Be familiar with the definitions relating to subspaces and basic facts about subspaces:
  • Definitions 3.1.1 (image), 3.1.2(span), 3.1.5(kernel), and Facts 3.1.3, 3.1.4, 3.1.6, and 3.1.7
  • Understand the terms domain and codomain (p. 106)
• Be familiar with the terminology and basic facts about subspaces, bases, and linear independence:
  • Definitions 3.2.1 (subspace), 3.2.3 (Redundancy, linear independence, basis) and 3.2.6 (linear relations)
  • Facts 3.2.2, 3.2.4, 3.2.5, 3.2.7, 3.2.8, and 3.2.10
  • Summary 3.2.9
• Understand the terminology and basic facts about dimension:
  • Definition 3.3.3, Facts 3.3.1, 3.3.2, 3.3.4, 3.3.6, 3.3.7, and 3.3.8
  • Algorithm 3.3.5 for constructing a basis of the image

Determinants

• Be familiar with the definitions relating to determinants and basic facts about them:
  • The Laplace expansion definition (Definition 6.1.4)
  • The permutations and inversions definition (Definition 6.2.9; Fact 6.2.10)
  • The Weierstrass definition (Problem 55 on page 274; Related website materials)
  • Sarrus's Rule for the determinant of a 3x3 matrix (Fact 6.1.2)
  • Minors (Definition 6.1.3); The Laplace expansion (Fact 6.1.5)
• Properties of determinants:
  • Facts 6.1.6, 6.1.7, 6.2.1, 6.2.2, 6.2.4, 6.2.6
  • Algorith 6.2.3 (Gauss-Jordan computation of determinant)
  • Cramer's Rule (Fact 6.3.9)
  • Classical adjoint (Fact 6.3.10)

Eigenvalues and Eigenvectors

• Be familiar with the definitions relating to eigenvalues and eigenvectors and basic facts about them:
  • Definitions 7.1.1 (eigenvalues and eigenvectors), 7.2.3 (trace), 7.2.6 (algebraic multiplicity), 7.3.1 (eigenspaces), 7.3.2 (geometric multiplicity), 7.3.3 (eigenbasis)
  • Facts 7.2.1 (characteristic equation), 7.2.2 (eigenvalues of a triangular matrix), 7.2.4 (characteristic equation of a 2x2 matrix), 7.2.5 (characteristic polynomial), 7.2.7(number of eigenvalues), 7.2.8 (eigenvalues, determinant, and trace) 7.3.4 (eigenbases and geometric multiplicity), 7.3.5 (nxn matrices with n distinct eigenvalues), 7.3.7 (geometric and algebraic multiplicity)

Chapter 4: Linear Spaces

Summary