LinearAlgebra03FallLetureNotes01

# LinearAlgebra03FallLetureNotes01 - HowardAnton ChrisRorres...

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Elementary Linear Algebra Howard Anton  Chris Rorres

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Chapter Contents 1.1 Introduction to  System of Linear           Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix Operations 1.4 Inverses; Rules of Matrix Arithmetic 1.5 Elementary Matrices and a Method for           Finding 1.6 Further Results on Systems of Equations           and Invertibility 1.7 Diagonal, Triangular, and Symmetric           Matrices  1 - A
1.1 Introduction to Systems of Equations

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Linear Equations Any straight line in xy-plane can be  represented algebraically by an equation of  the form: General form: define a  linear equation  in the  n   variables                   : Where                        and b are real constants. The variables in a linear equation are sometimes          called  unknowns .      b y a x a = + 2 1 n x x x ,..., , 2 1 b x a x a x a n n = + + + ... 2 2 1 1 , ,..., , 2 1 n a a a
Example 1 Linear Equations The equations                                      and                               are linear.    Observe that a linear equation does not involve any  products or roots of variables. All variables occur only to  the first power and do not appear as arguments for  trigonometric, logarithmic, or exponential functions.  The equations      are  not  linear. solution  of a linear equation is a sequence of n numbers                                                   such that the equation is satisfied. The set of  all solutions of the equation is called its  solution set  or  general solution  of the equation , 1 3 2 1 , 7 3 + + = = + z x y y x 7 3 2 4 3 2 1 = + - - x x x x x y xz z y x y x sin and , 4 2 3 , 5 3 = = + - + = + n s s s ,..., , 2 1

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Example 2 Finding a Solution Set (1/2) Find the solution of Solution(a)       we can assign an arbitrary value to x and solve for y ,  or choose an arbitrary value for y and solve for x .If we  follow the first approach and assign x an arbitrary  value ,we obtain                      arbitrary numbers          are called   parameter . for example  1 2 4 ) a ( = - y x 2 2 1 1 , 4 1 2 1 or 2 1 2 , t y t x t y t x = + = - = = 2 , 1 t t 2 11 as 2 11 , 3 solution the yields 3 2 1 = = = = t y x t
Example 2 Finding a Solution Set (2/2) Find the solution of Solution(b)       we can assign arbitrary values to any two  variables and solve for the third variable.

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