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Math 221, Section 1.1 Today: (1) setting up terminologies, notations, ideas, ... (2) solve a system of linear equations 1 / 11 Linear equations A linear equation in n variables/unknowns x1 , . . . , xn is an equation that can be written in the form a1 x1 + · · · + an xn = b where the coefficients a1 , . . . , an and the constant term b are known in advance. Examples equation in 2 variables ax + by = c equation in 3 variables ax + by + cz = d. 2 / 11 System of linear equations A system of linear equations (or a linear system) is a collection of one or more linear equations. Examples in the book I 2x − y + 1.5z = 8 x − 4z = −7 I x − 2y + z = 0 2y − 8z = 8 −4x + 5y + 9z = 9 3 / 11 Solutions I A solution of the system is an array (s1 , . . . , sn ) of numbers that makes each equation a true statement when the values (s1 , . . . , sn ) are substituted for the variables (x1 , . . . , xn ) respectively. I The set of all possible solutions is called the solution set of the linear system. I 3 situations happen: exactly one/unique solution, no solution, or infinitely many solutions 4 / 11 Solutions An example in the book 2x − y + 1.5z = 8 x − 4z = −7 I has a solution (−7, −22, 0), I but the solution set is (−7 + 4t, −22 + 9.5t, t), where t is any real number. 5 / 11 Matrix Notation Write the system x − 2y + z = 0 2y − 8z = 8 −4x + 5y + 9z = 9 as an augmented matrix 1 −2 1 0 0 2 −8 8 −4 5 9 9 [coefficient matrix | constant vector] 6 / 11 Solving a Linear System Solving the system by elimination x − 2y + z = 0 2y − 8z = 8 −4x + 5y + 9z = 9 VS Reducing the matrix by row operations 1 −2 1 0 0 2 −8 8 −4 5 9 9 7 / 11 8 / 11 Elementary row operations What kind of operations did we apply to a matrix? 3 elementary row operations I (Replacement) Add to one row a multiple of another row. I (Scaling) Multiply all entries in a row by a nonzero constant. I (Interchanging) Interchange two rows. 9 / 11 Consistency A system of linear equations is said to be consistent/solvable if it has either one solution or infinitely many solutions. A system is inconsistent/unsolvable if it has no solution. Examples 10 / 11 Consistency Examples 11 / 11