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

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## A.G. Hamilton:

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

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

- ISBN 0-521-31041-5

### Subject Matter

### 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.

*Answers to exercises**Sample test papers**Further reading**Index*

## Source work progress

- 1987: A.G. Hamilton:
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