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### 21 Cards in this Set

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 A linear equation in n variables x1, x2, x3, ... , xn has the form __________. a1x1 + a2x2 + a3x3 + ... + anxn = b A systems of linear equations in which each of the constant terms is zero is called ________. Homogeneous Reduced row echelon form (rref): 1. all rows consisting entirely of zeros at bottom 2. the first nonzero entry in any nonzero row is 1 (called leading 1) 3. the leading in each nonzero row is at least one column to right of leading 1 in previous row 4. all entries ABOVE (and below) any leading 1 are zeros To solve a system in ref you use a procedure called __________. back substitution Two systems of linear equations are called ________ if they have precisely the same solution set. Equivalent Operations that lead to equivalent systems of equations. For a system of linear equations in n variables, precisely one of the following is true: 1. The system has exactly one solution (consistent system). 2. The system has an infinite number of solutions (consistent system). 3. The system has no solution (inconsistent system). The number a1 is the ________ coefficient. Leading Elementary Row Operations (ERO): Elementary Row Operations (ERO): 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row. If all variables in a homogeneous system have the value zero, then each of the equations must be satisfied. Such a solution is called _______ (or obvious). Trivial An m*n matrix (read "m by n") has m _____ (horizontal lines) and n _______ (vertical lines). rows, columns If m and n are positive integers, then an m*n ______ is a rectangular array in which each entry, aij, of the matrix is a number. Matrix A homogeneous system must have at least ___________. one solution A system of linear equations is called _________ if it has at least one solution and _________ if it has no solution. consistent, inconsistent A matrix in row-echelon form (ref) has the following properties: 1. All rows consisting entirely of zeros occur at the bottom of the matrix. 2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). 3. For two succesive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. x1 is the leading _______. Variable For a system of linear equations in n variables, precisely one of the following is true: 1. The system has exactly one solution (consistent system). 2. The system has an infinite number of solutions (consistent system). 3. The system has no solution (inconsistent system). The ______ term b is a real number. Constant The real numbers (a1, a2, a3, ... , an) are ________. Coefficients Row-echelon form (ref) means: the system has leading coefficients of 1 that follows a stair-step pattern example: 1 2 3 9 0 1 3 5 0 0 1 2 The set of all solutions of a linear equations is called its __________, and when this set is found, the equation is said to have been __________. solution set, solved Every homogeneous system of linear equations is __________. Moreover, if the system has fewer equations than variables, then it must have an __________ number of solutions. consistent, infinite