Lin-Alg - Part I Linear Algebra 1.1 Linear Equations Some...

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Part I. Linear Algebra 1.1 Linear Equations; Some Geometry A linear (algebraic) equation in n unknowns , x 1 ,x 2 ,. . . n , is an equation of the form a 1 x 1 + a 2 x 2 + ··· + a n x n = b where a 1 ,a 2 . . n and b are given numbers called the coefficients . In particular ax = b is a linear equation in one unknown; ax + by = c is a linear equation in two unknowns (if a and b are real numbers, not both 0, then the graph of the equation is a straight line); and ax + by + cz = d is a linear equation in three unknowns (if a, b and c are real numbers, not all 0, then the graph is a plane in 3-space). Our main interest in this treatment of linear algebra is solving systems of linear equations. Remark: In a general study of Linear Algebra the coefficients and the values of the unknowns are assumed to come from some given Feld F . In the treatment here we will use the Feld of real numbers, R ; the term “number” means “real number.” ± Linear equations in one unknown. We begin with simplest case: one equation in one unknown. If you were asked to Fnd a real number x such that ax = b you would probably say “that’s easy,” x = b a . But the fact is, this “solution” is not necessarily correct. ±or example, consider the three equations (1) 2 x =6 , (2) 0 x , (3) 0 x =0 . ±or equation (1), the solution x / 2 = 3 is correct. However, consider equation (2); there is no real number that satisFes this equation! Now look at equation (3); every real number satisFes (3). In general, it is easy to see that for the equation ax = b , exactly one of three things happens: (a) There is precisely one solution ( x = b/a , when a ± = 0). (b) There are no solutions ( a ,b ± = 0). 1
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(c) There are infnitely many solutions ( a = b = 0). As you will see, this simple case illustrates (and motivates) the general situation. For any system of m linear equations in n unknowns, exactly one of three possibilities occurs: a unique solution, no solution, or in±nitely many solutions . Linear equations in two unknowns We begin with the one equation: ax + by = c. Here we are looking For ordered pairs oF real numbers ( x,y ) which satisFy the equation. IF a = b =0 and c ± = 0, then there are no solutions. IF a = b = c = 0, then every ordered pair ( ) satisfes the equation; there are infnitely many solutions, the whole xy -plane, a two-dimensional set. IF at least one oF a and b is different From 0, then the equation ax + by = c represents a straight line in the xy -plane and the equation has infnitely many solutions, the set oF all points on the line, this time a one-dimensional set. Note that For one linear equation in two unknowns it is not possible to have a unique solution; we either have no solution or infnitely many solutions. Two linear equations in two unknowns is a more interesting case. IF a and b are not both zero, and c and d are not both zero, then the pair oF equations ax + by = α cx + dy = β represents a pair oF lines in the xy -plane. We are looking For ordered pairs ( ) oF numbers that satisFy both equations simultaneously. As you know, two lines in the plane either (a) have a unique point oF intersection (this occurs when the lines have different slopes), or
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Lin-Alg - Part I Linear Algebra 1.1 Linear Equations Some...

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