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Unformatted text preview: Elementary Linear Algebra Howard Anton  Chris Rorres 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 Linear Equations Any straight line in xyplane 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. A 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 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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This note was uploaded on 01/12/2011 for the course MAT/116 mat/116 taught by Professor Unknown during the Spring '10 term at University of Phoenix.
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