## Basic Linear AlgebraBasic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers: this will take the form of a tutorial on the use of the "LinearAlgebra" package in MAPLE 7 and will deal with all the aspects of linear algebra developed within the book. |

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### Contents

The Algebra of Matrices | 1 |

Some Applications of Matrices | 17 |

Systems of Linear Equations | 27 |

Invertible Matrices | 59 |

Vector Spaces | 69 |

Linear Mappings | 95 |

The Matrix Connection | 113 |

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### Common terms and phrases

algebraic multiplicity ASSIGNMENT EXERCISES augmented matrix basis of IR3 characteristic polynomial coefficient matrix column operations column rank complex numbers compute Consider the matrix Corollary deduce defined Definition denote determinantal mapping diagonalisable dim Ker eigenvalues eigenvectors elementary matrices elementary row operations elements end if end end proc Example expressed field F finite-dimensional vector space functions Hence Hermite form i-th row induction injective invertible matrix IRn[X isomorphism Laplace expansion linear combination linear equations linear mapping linearly independent linearly independent subset m x n mA(X Matnxn F minimum polynomial multiplication by scalars n x n matrix natural ordered basis non-zero rows normal form observe obtain orthogonal permutation positive integer real number real vector space relative right inverse row and column row rank row-echelon form row-equivalent Show skew-symmetric spanning set square matrix subspace Suppose surjective system of equations Theorem 5.8 unique whence zero