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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
