REF -- Row Echelon Form 2 5 2

Solving linear systems

Math 40, Introduction to Linear Algebra January 2012

REF -- row echelon form

25 2 7 Row reduced matrix from cereal example: 0 7 3 8  − −405 −135  00 7 7 A matrix is in row   echelon form (REF) if it satisﬁes the following: Is REF of a matrix unique? NO! • any all-zero rows are at

the bottom ﬁrst nonzero entries in each row

• leading entries form a staircase pattern more formally each leading entry is in a column to the right of the leading entry above it Gaussian elimination

Given a linear system, the process of

• expressing it as an augmented Can we accomplish matrix, backwards substitution via EROs done after • performing EROs on the matrix is in REF? augmented matrix to get it in REF,

• and, ﬁnally, using back substitution to solve the system YES! is called Gaussian elimination.

Example 1

x1 + x2 + x3 =2 1112 2x1 +3x2 + x3 =3 2313  1 1 2 6  x1 x2 2x3 = 6 − − − − − −   rewrite linear system as 1112 augmented matrix R2 2R1 − 0 1 1 1  − −  R3 R1 0 2 3 8 − − − −   11 1 2 01 1 1  − −  R3 +2R2 0 0 5 10 − −   matrix is in REF

We could now convert to back to equations and do back substitution... Example 1

x1 + x2 + x3 =2 11 1 2 01 1 1 2x +3x + x =3 1 2 3  00−5 −10  x x 2x = 6 − − 1 − 2 − 3 −   rewrite linear system as 11 1 2 augmented matrix 1 01 1 1 R  − −  −5 3 00 1 2 1112   2313  1 1 2 6  1100 − − − R1 R3   − 1112 0101 R2 2R1 R + R − 0 1 1 1 2 3    − −  0012 R3 R1 0 2 3 8 − − − −   x1 x2 x3  11 1 2 R R 100 1 x1 1 01 1 1 1 2 − −  − −  − 010 1 x = x = 1 R3 +2R2 0 0 5 10 2 − −         001 2 x3 2 matrix is in REF      

RREF -- reduced row echelon form

A matrix is in reduced row echelon form (RREF) if it 100 1 satisﬁes the following: 010−1   • any all-zero rows are at the 001 2 bottom  

• leading entries form each leading entry is in a column to staircase pattern more formally the right of the leading entry above it

• leading entries are ones Is RREF of a • each column with a leading matrix unique? YES! one has zeros elsewhere RREF -- reduced row echelon form

A matrix is in reduced row echelon form (RREF) if it Example: satisﬁes the following: 10001 00 0 00101 0 • any all-zero rows are at the 00011 0 bottom 0000011    00000000 • leading entries form   00000000 staircase pattern     • leading entries are ones Where must there be zeros, given leading • each column with a leading entries above? one has zeros elsewhere

Gauss-Jordan elimination

Given a linear system, the process of

• expressing it as an augmented matrix,

• performing EROs on the augmented matrix to get it in RREF,

• and writing the solution to the system

is called Gauss-Jordan elimination. In REF, RREF, or none of the above?

10 90 4 01− 3 0 1 241 00 0 1−7 0 12 −  −  00 0 0 1 000     121 3  REF  REF 000− 0 000 0   1 RREF 1 5 01− 123 023 REF 101 1000 1100 REF 000   010 0110 0011 none of   none of the the above   above

Example 2

2x 6y =1 2 6 1 − − x +3y = 1 131 − − − −

2 6 1 1 − 1 R2 + R1 00 2 − 2

2x 6y =1 − System is inconsistent 1 0= no solutions −2

Geometrically, this means the two lines are parallel. Example 3

x1 +2x2 +3x3 =4 1234 5x +6x +7x =8 5678 1 2 3 123 4 R2 5R1 0 4 8 12 − − − − 1234 1 R2 0123 −4 R 2R 10 1 2 1 − 2 01− 2−3 matrix is in RREF

Example 3

x1 x2 x3 x1 +2x2 +3x3 =4 10 1 2 01− 2−3 5x1 +6x2 +7x3 =8 x1,x2 are leading variables x3 is a free variable • columns w/leading entries correspond to x1 x3 = 2 x1 = x3 2 − − − leading variables x +2x =3 x = 2x +3 2 3 2 − 3 x3 is free x3 is free • columns without leading entries x1 x3 2 1 2 − − correspond to x = x2 = 2x3 +3 = x3 2 + 3 free variables   −  −    x3 x3 1 0

    wherex3 is any  system has inﬁnite real number # of solutions 3 line in R Example 4

2x + x +4x =4 − 1 2 3 3 3x x 8x = 12 1 − 2 2 − 3 − 2x x 6x = 10 1 − 2 − 3 − 21 4 4 1 1 0 4 − EROs − 2 3 3 8 12 0013 RREF  − 2 − −    2 1 6 10 0000 − − −   x1,x3 are leading variables 1 x2 is a free variable x = x +4 1 2 2 1 1 2 x2 +4 2 4 for any x2 is free x = x2 = x2 1 + 0       x2 R x3 =3 3 0 3 ∈      

Recognizing the size of solution set

represents a Suppose augmented matrix is in RREF. nonzero value

(1) If any row is of the form 00 0 ··· then the system is inconsistent and there are no solutions.

(2) If the system is consistent and if there exist free variables, then there are an inﬁnite number of solutions. What does a row of all zeros (3) Otherwise, system has a unique in RREF mean? solution.