Linear Algebra 1.1 & 1.2 definitions

system of equations

a collection of one or more linear equations with the same

variables

solution

a list (s1, s2, . . . , sn) of real numbers that make each equation a true statement when the values s1, . . . , sn are substituted for x1, . . . xn, respectively.

solution set

set of all possible solutions

equilvalent

Two linear systems are ________ if they have the same solution set.

linear equation

a1x2 + a2x2 + · · · + anxn = b,

consistent

a(n) ______ system has either one or infinitely many solutions

inconsistent

a(n) ______ system has no solutions

m rows and n columns

an m x n matrix has …

1. Add a multiple of one equation to another.

2. Interchange two equations.

3. Multiply an equation by a nonzero constant.

3 basic row operations

1. Replacement

2. Interchange

3. Scaling

Elementary Row Operations

row equilvalent ~

two matrices are ________ when one can be obtained by the other with elementary row operations

• All rows of zeroes are at the bottom

• The leading entry in a row is in a column to the right of the leading entry of the row

above it.

• All entries in a column below a leading entry are zeroes

Row Echelon Form

• Each leading entry in a nonzero row is 1

• Each leading 1 is the only nonzero entry in its column.

Reduced Row Echelon Form

echelon form; reduced echelon form

If a matrix A can be reduced to a matrix U in REF, we call U a(n) ______ of A; if U is in RREF we call U the ________ of A.

F

There are many reduced row matrices that a matrix can be row equivalent to T/F

pivot

a _____ position in a matrix A is a location in A that corresponds to a

leading 1 in the reduced echelon form of A. A _____ column is a column of A that contains a ______ position.

basic

a ____ variable corresponds to a pivot column of the augmented matrix

free

a ____ variable correspond to non-pivot columns.

the rightmost column is not a pivot column

a matrix is consistent iff … that is if the echelon form of the augmented matrix has no row of the form [0 . . . 0 b], with b nonzero