The number of free variables is called the dimension of the solution set. The above examples show us the following pattern: when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc. There is a natural relationship between the number of free variables and the “size” of the solution set, as follows. There is a natural question to ask here: is it possible to write the solution to a homogeneous matrix equation using fewer vectors than the one given in the above recipe? We will see in example in Section 2.5 that the answer is no: the vectors from the recipe are always linearly independent, which means that there is no way to write the solution with fewer vectors.Īnother natural question is: are the solution sets for inhomogeneuous equations also spans? As we will see shortly, they are never spans, but they are closely related to spans. Since two of the variables were free, the solution set is a plane. Since there were three variables in the above example, the solution set is a subset of R 3. ![]() The equation Ax = 0 has a nontrivial solution ⇐⇒ there is a free variable ⇐⇒ A has a column without a pivot position. Any nonzero solution is called nontrivial. DefinitionĪ system of linear equations of the form Ax = 0 is called homogeneous.Ī system of linear equations of the form Ax = b for b B = 0 is called inhomogeneous.Ī homogeneous system is just a system of linear equations where all constants on the right side of the equals sign are zero.Ī homogeneous system always has the solution x = 0. The equation Ax = b is easier to solve when b = 0, so we start with this case. In this section we will study the geometry of the solution set of any matrix equation Ax = b. Vocabulary words: homogeneous/ inhomogeneous, trivial solution.Pictures: solution set of a homogeneous system, solution set of an inhomogeneous system, the relationship between the two.Recipes: parametric vector form, write the solution set of a homogeneous system as a span.Understand the difference between the solution set and the column span.Understand the relationship between the solution set of Ax = 0 and the solution set of Ax = b. ![]() Hints and Solutions to Selected Exercises.Zheng, Value Distribution of Meromorphic Functions, Tsinghua University Press (Beijing, 2010).3 Linear Transformations and Matrix Algebra Yang, On Petrenko’s deviations and Julia limiting directions of solutions of complex differential equations, J. Yang, Radial distribution of Julia sets of derivatives of solutions to complex linear differential equations, Sci. Yang, Value Distribution Theory, Springer-Verlag (Berlin, 1993). Qiu, Second-order complex linear differential equations with special functions or extremal functions as coefficients, Electron J. Zhang, Julia limiting directions of entire solutions of complex differential equations, Acta. Yao, On Julia limiting directions of meromorphic functions, Israel J. Rudin, Real and Complex Analysis, McGraw-Hill (New York, 1987). Qiao, On limiting directions of Julia sets, Ann. Qiao, Stable sets for iterations of entire functions, Acta. Petrenko, Growth of meromorphic functions of finite lower order, Izv. ![]() Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter (Berlin, 1993). Ye, Lower order and Baker wandering domains of solutions to differential equations with coeffidicnets of exponential growth, J. Jackson, On q-difference equations, Amer. Wang, On limit directions of Julia sets of entire solutions of linear dffierential euqation, J. ![]() Wang, On the radial distribution of Julia sets of entire solutions of f ( n) + A( z) f = 0, J. Zemirni, On Petrenko’s deviations and second order differential equations, Kodai Math. Hayman, Meromorphic Functions, Clarendon Press (Oxford, 1964). Ostrowskii, Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs Series, Amer. Edrei, Sums of deficiencies of meromorphic functions, J. Wu, Radial distribution of Julia sets of entire solutions to complex difference equations, Mediterr. Wang, Nevanlinna theory for Jackson difference operators and entire solutions of q-difference equations, Anal. Baker, Sets of non-normality in iteration theory, J. Baernstein, Proof of Edrei’s spread conjecture, Proc.
0 Comments
Leave a Reply. |