Analytical Geometry and Linear Algebra
Authors
Description
This is an introduction to Analytic Geometry, i.e. the use of coordinates to study Euclidean Geometry – flat and spatial.
As and when necessary, vectors are introduced, taking advantage of their notational simplicity and strong geometric appeal.
Linear systems are shown as an example of connecting algebra with geometry, motivating the consideration of matrices and the linear dependence between their rows and columns. Areas and volumes lead to the study of determinants. Conics and quadrics lead to quadratic forms, symmetric matrices and their eigenvalues.
In general, the book shows how basic Linear Algebra concepts are useful for treating Analytic Geometry problems efficiently and elegantly.
Target audience
Higher education
Name: Analytical Geometry and Linear Algebra
Author(s): e Elon Lages Lima
Pages: 324
Publication: IMPA, 2015
ISBN: 978-85-244-0462-7
Edition: 2
Introduction
1 Coordinates on the line
2 Coordinates in the plane
3 Line segments in the plane
4 The distance between two points
5 Choosing the coordinate system
6 Other types of coordinates
7 The equations of the line
8 Angle between two lines
9 Distance from a point to a line
10 Area of a triangle
11 Linear inequalities
12 Equation of the circle
13 Recognizing the equation of a circle
14 Vectors in the plane
15 Operations with vectors
16 Equation of the ellipse
17 Equation of the hyperbola
18 Equation of the parabola
19 Change of coordinates
20 Quadratic forms
21 The general equation of the second degree
22 The sign of a quadratic form
23 Linear transformations
24 Coordinates in space
25 The parametric equations of a line
26 Distance between two points in space
27 Line segments in space
28 Vectors in space
29 Equation of the plane
30 Systems of linear equations with two unknowns
31 Systems of linear equations with three unknowns
32 Three linear equations with three unknowns
33 Scaling (Gaussian elimination)
34 Operations with matrices
35 Determinants
36 Cramer’s rule
37 The determinant of the product of two matrices
38 Areas, volumes and the Gram matrix
39 Characterization of invertible matrices
40 The vector product
41 Changing coordinates in space
42 Quadratic forms in R 3
43 Central quadratics
44 Completing squares in R 3
45 The general equation of the second degree in R 3
46 Matrices and quadratic forms
47 Linear transformations in R3
Bibliography