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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