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

Subject Matter

 * Linear Algebra

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
* : $1$: Gaussian Elimination