section12 (2) - Linear Equations and Matrices MAT188H1F...

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Unformatted text preview: Linear Equations and Matrices MAT188H1F Lec03 Burbulla Chapter 1 Lecture Notes Fall 2007 Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Linear Equations and Matrices Linear Equations and Matrices Linear Equations Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Linear Equations and Matrices Linear Equations What Is A Linear Equation? A linear equation in variables x 1 , x 2 , x 3 , . . . , x n is an equation of the form a i 1 x 1 + a i 2 x 2 + a i 3 x 3 + + a in x n = b i , where a i 1 , a i 2 , a i 3 , . . . , a in and b i are all scalars (ie. real numbers.) Note that each variable is first power only; linear equations do not include I higher powers of variables, or products of variables, I negative powers of variables, I exponential or logarithmic functions of a variable, I or any trigonometric functions of a variable. Linear equations are simple! Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Linear Equations and Matrices Linear Equations What Is A System of Linear Equatons? m linear equations in n variables form a system of linear equations. The scalars a ij , for 1 i m , 1 j n are called the coefficients of the system. The numbers b i for 1 i m are called the constants of the system. If n = 2 , we offten use x and y for the two variables. If n = 3 , we often use x , y and z for the three variables. The simplest system of all is one equation in one variable: ax = b . The solution of this system is x = b a if a 6 = 0 . If a = 0 and b = 0 , then x can be any real number; there are infinitely many solutions. But if a = 0 and b 6 = 0 , then the equation has no solutions at all. Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Linear Equations and Matrices Linear Equations The General System of Two Equations in Two Variables This system can be written as ax + by = p cx + dy = q You should recognize from high school that geometrically the solution to this system represents the intersection of two lines in the plane. The possibilities are I the two lines intersect in exactly one point I the two lines are parallel and distinct, so there is no intersection point I the two lines are coincident, so that the solution set is the whole line Examples follow. Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Linear Equations and Matrices Linear Equations Example 1: A Unique Intersection Point Consider the system x + y = 7 x- y = 1 Adding the two equations gives 2 x = 8 x = 4 . Substituting this value of x into either of the original equations allows you to solve for y and find that y = 3 . Thus the only intersection point of these two lines is ( x , y ) = (4 , 3) . Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Linear Equations and Matrices Linear Equations Example 2: Two Parallel Lines Consider the system x + y = 7 2 x + 2 y = 10 In this case, if you attempt to eliminate a variable, say x , by doubling the first equation and subtracting the second, you obtain x + 0 y = 4 0 = 4 , which is impossible no matter what x...
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section12 (2) - Linear Equations and Matrices MAT188H1F...

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