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A First Course In Linear Algebra


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Table of Contents

* Introduction *1 Geometric Vectors *1.1 Addition of geometric vectors *1.2 Multiplication by a scalar *1.3 Subtraction of vectors *1.4 List of useful properties *1.5 The geometry of parallelograms *2 Position Vectors and Components *2.1 Magnitude, unit vectors and hat notation *2.2 Parallel vectors *2.3 Position vectors and components *2.4 Length of a vector *2.5 Linear independence for two vectors *3 Dot Products and Projections *3.1 Geometric definition of dot product *3.2 Algebraic definition of dot product *3.3 Angle between two vectors *3.4 Projections and orthogonal components *3.5 Another application to geometry in the plane *4 Cross Products *4.1 Definition of cross product *4.2 List of useful properties *4.3 Method of expanding brackets *4.4 Geometric interpretation *4.5 Continuity and the right-hand orientation *5 Lines in Space *5.1 Parametric vector and scalar equations of a line *5.2 Cartesian equations of a line *5.3 Finding a line using two points *5.4 Distance from a point to a line *6 Planes in Space *6.1 Vector equation of a plane *6.2 Cartesian equation of a plane *6.3 Finding a plane using three points *6.4 Distance from a point to a plane *7 Systems of Linear Equations *7.1 Consistent and inconsistent systems *7.2 Parametric solutions *7.3 Augmented matrix of a system *7.4 Gaussian elimination *7.5 Reduced row echelon form *8 Matrix Operations *8.1 Addition, subtraction and scalar multiplication *8.2 Matrix multiplication *8.3 Connections with systems of equations *9 Matrix Inverses *9.1 Identity matrices and inverses *9.2 Inverses of two-by-two matrices *9.3 Powers of a matrix *9.4 Using row reduction to find the inverse *9.5 Using inverses to solve systems of equations *9.6 Elementary matrices *10 Determinants *10.1 Determinant of a 3 x 3 matrix *10.2 Cross products revisited *10.3 Properties of determinants *10.4 Orientation of a triangle *11 Eigenvalues and Eigenvectors *11.1 Existence of eigenvalues *11.2 Finding eigenvalues *11.3 Reflections and rotations in the plane *12 Diagonalising a Matrix *12.1 An example which cannot be diagonalised *12.2 An example of a Markov process *12.3 The Jordan form of a matrix * Hints and Solutions * Appendix 1 The Theorem of Pythagoras * Appendix 2 Mathematical Implication * Appendix 3 Complex Numbers

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