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### 29 Cards in this Set

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 Linear Equation An equation that can be written in the form a1x1 + a2x2 + …+ anxn = b, a,b = number x = unknowns coefficients real or complex numbers system of linear equations (linear system) a collection of one or more linear equations involving the same variables solution a list (s1, s2,..., sn) of numbers that makes each equation a true statement when the values s1...sn are substituted for x1...xn. solution set The set of all possible solutions Equivalent Linear systems are equivalent if they have the same solution set Consistent A linear system is consistent if it has either one solution or infinitely many solutions Inconsistent A linear system is inconsistent if it has no solution Matrix A rectangular array of numbers Coefficient matrix A matrix who's entries are the coefficients of a linear system Augmented matrix A matrix that contains the coefficients and constants of a linear system Size of matrix Tells how many rows and columns a matrix has m x n matrix m = number of rows n = number of columns Elementary Row Operations 1. Replacement 2. Interchange 3. Scaling Replacement Replace one row by the sum of itself and a multiple of another row Interchange Interchange two rows Scaling Multiple all entries in a row by a nonzero constant Leading Entry leftmost nonzero entry in a nonzero row Echelon form 1. All nonzero rows are above any rows of all zeros 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. Reduced echelon form 3. All leading entries are 1 4. Each leading entry is the only nonzero in its column Row Reduced Transformed by elementary row ops Pivot Position Corresponds to a leading entry in the reduced echelon form Pivot is a nonzero number in a pivot position that is used as needed to create zeros via row operations Forward Phase The process that results in an echelon form Backward Phase Last step that turns an echelon form in to a reduced echelon form. Partial Pivoting Selecting the pivot with the largest absolute value, reduces roundoff errors in computers. Basic Variable A variable that corresponds to a pivot column in the coefficient matrix Free variable Any variable in a linear system that isn't a basic variable General Solution A description of all solutions