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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
