# Book:A.G. Hamilton/A First Course in Linear Algebra

## A.G. Hamilton: A First Course in Linear Algebra with Concurrent Examples

Published $\text {1987}$, Cambridge University Press

ISBN 0-521-31041-5

### Contents

Preface
1 Gaussian elimination
Description and application of an algorithm to reduce a matrix to row echelon form. Partial pivoting.
2 Solutions to simultaneous equations 1
Use of the GE algorithm. The different possible outcomes. Inconsistent equations. Solutions involving arbitrary parameters.
3 Matrices and algebraic vectors
Sums and producs of matrices. Algebraic laws. Simultaneous linear equations considered as a single matrix equation.
4 Special matrices
Zero matrix, diagonal matrices, identity matrix, triangular matrices. Transpose of a matrix, symmetric and skew-symmetric matrices. Elementary matrices and their relation with elementary row operations.
5 Matrix inverses
Invertible and singular matrices. Algorithm for finding inverses. Inverses of products.
6 Linear independence and rank
Algorithms for testing linear dependence or independence. Rank of a matrix. Equivalence of invertibility with conditions involving rank, linear independence and solutions to equations (via the GE algorithm).
7 Determinants
$2 \times 2$ and $3 \times 3$ determinants. Methods for evaluation. Effects of elementary row operations. A matrix is invertible if and only if its determinant is non-zero. Determinant of a product. Adjoint matrix. Indication of extension to larger determinants.
8 Solutions to simultaneous equations 2
Rules involving the ranks of matrices of coefficients and whether the matrix is invertible.
9 Vectors in geometry
Representing vectors by directed line segments. Algebraic operations interpreted geometrically. The Section Formula. The standard basis vectors $i$, $j$, $k$. The length of a vector.
10 Straight lines and planes
Straight lines using vector equations. Direction ratios. Scalar product of two vectors. Angles between lines. Planes. Intersections of planes.
11 Cross product
Definition and properties of the vector product. Areas and volumes. Scalar triple product. Coplanar vectors. Link with linear dependence via determinants.