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