tailieunhanh - CS 205 Mathematical Methods for Robotics and Vision - Chapter 2

Algebraic Linear Systems Một hệ thống đại số tuyến tính là một tập hợp các phương trình m trong n vô hướng không rõ, xuất hiện tuyến tính. Nếu không có mất tính tổng quát, một hệ thống đại số tuyến tính có thể được viết như sau: | Chapter 2 Algebraic Linear Systems An algebraic linear system is a set of m equations inn unknown scalars which appear linearly. Without loss of generality an algebraic linear system can be written as follows Ax b where A is an m X n matrix x is an n -dimensional vector that collects all of the unknowns and b is a known vector of dimension m. In this chapter we only consider the cases in which the entries oL4 b and x are real numbers. Two reasons are usually offered for the importance of linear systems. The first is apparently deep and refers to the principle of superposition of effects. For instance in dynamics superposition of forces states that if force f produces acceleration a both possibly vectors and force f2 produces acceleration a then the combined force f of produces acceleration a a2. This is Newton s second law of dynamics although in a formulation less common than the equivalent f ma. Because Newton s laws are at the basis of the entire edifice of Mechanics linearity appears to be a fundamental principle of Nature. However like all physical laws Newton s second law is an abstraction and ignores viscosity friction turbulence and other nonlinear effects. Linearity then is perhaps more in the physicist s mind than in reality if nonlinear effects can be ignored physical phenomena are linear A more pragmatic explanation is that linear systems are the only ones we know how to solve in general. This argument which is apparently more shallow than the previous one is actually rather important. Here is why. Given two algebraic equations in two variables f X y 9 x y we can eliminate say t and obtain the equivalent system F X y h X . Thus the original system is as hard to solve as it is to find the roots of the polynomial in a single variable. Unfortunately if f and g have degrees dy and dg the polynomial has generically degree dfdg. Thus the degree of a system of equations is roughly speaking the product of the degrees. For instance a system of m quadratic .