First Course in Linear Algebra, A, Pearson Original Edition, 3rd edition

Published by Pearson (December 15, 2010) © 2011

  • David Easdown The University of Sydney
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Title overview

An engaging introductory text to linear algebra for new students entering university and returning mature-age students. It aims to make critical algebraic concepts easy to understand.

Table of contents

  • Introduction
  • 1 Geometric Vectors
  • 1.1 Addition of geometric vectors
  • 1.2 Multiplication by a scalar
  • 1.3 Subtraction of vectors
  • 1.4 List of useful properties
  • 1.5 The geometry of parallelograms
  • 2 Position Vectors and Components
  • 2.1 Magnitude, unit vectors and hat notation
  • 2.2 Parallel vectors
  • 2.3 Position vectors and components
  • 2.4 Length of a vector
  • 2.5 Linear independence for two vectors
  • 3 Dot Products and Projections
  • 3.1 Geometric definition of dot product
  • 3.2 Algebraic definition of dot product
  • 3.3 Angle between two vectors
  • 3.4 Projections and orthogonal components
  • 3.5 Another application to geometry in the plane
  • 4 Cross Products
  • 4.1 Definition of cross product
  • 4.2 List of useful properties
  • 4.3 Method of expanding brackets
  • 4.4 Geometric interpretation
  • 4.5 Continuity and the right-hand orientation
  • 5 Lines in Space
  • 5.1 Parametric vector and scalar equations of a line
  • 5.2 Cartesian equations of a line
  • 5.3 Finding a line using two points
  • 5.4 Distance from a point to a line
  • 6 Planes in Space
  • 6.1 Vector equation of a plane
  • 6.2 Cartesian equation of a plane
  • 6.3 Finding a plane using three points
  • 6.4 Distance from a point to a plane
  • 7 Systems of Linear Equations
  • 7.1 Consistent and inconsistent systems
  • 7.2 Parametric solutions
  • 7.3 Augmented matrix of a system
  • 7.4 Gaussian elimination
  • 7.5 Reduced row echelon form
  • 8 Matrix Operations
  • 8.1 Addition, subtraction and scalar multiplication
  • 8.2 Matrix multiplication
  • 8.3 Connections with systems of equations
  • 9 Matrix Inverses
  • 9.1 Identity matrices and inverses
  • 9.2 Inverses of two-by-two matrices
  • 9.3 Powers of a matrix
  • 9.4 Using row reduction to find the inverse
  • 9.5 Using inverses to solve systems of equations
  • 9.6 Elementary matrices
  • 10 Determinants
  • 10.1 Determinant of a 3 × 3 matrix
  • 10.2 Cross products revisited
  • 10.3 Properties of determinants
  • 10.4 Orientation of a triangle
  • 11 Eigenvalues and Eigenvectors
  • 11.1 Existence of eigenvalues
  • 11.2 Finding eigenvalues
  • 11.3 Reflections and rotations in the plane
  • 12 Diagonalising a Matrix
  • 12.1 An example which cannot be diagonalised
  • 12.2 An example of a Markov process
  • 12.3 The Jordan form of a matrix
  • Hints and Solutions
  • Appendix 1 The Theorem of Pythagoras
  • Appendix 2 Mathematical Implication
  • Appendix 3 Complex Numbers

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