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1.2 Gaussian Elimination P. Danziger

mEquations innUnknowns

Givennvariablesx1,x2;:::;xnandn+1 constants

a

1,a2;:::;an;bthe equation

a

1x1+a2x2+:::+anxn=b

represents ann1 dimensional object inn-space, called a hyperplane.

We want to consider the situation where we have

msuch equations a

11x1+a12x2+:::+a1nxn=b1

a21x1+a22x2+:::+a2nxn=b2...... a m1x1+am2x2+:::+amnxn=bm

This is called a system ofm(linear) equations in

nunknowns (or variables).

We want to nd solutions of this system of equa-

tions. 1

1.2 Gaussian Elimination P. Danziger

Theorem 1Given a system ofmequations inn

unknowns:

Ifm < nthen the number of parameters in the

solution will beat leastnm. (Thus if there is a unique solution we must havemn.)

Ifm > nthe system is calledoverprescribed.

Overprescribed systems either have no solu-

tion or they contain reduncancy. redundancy means that we can nd (mn) equations which can be dropped without aecting the solution. If a system of equations has no solution it is called inconsistent If a system of equations has at least one solution it is calledconsistent 2

1.2 Gaussian Elimination P. Danziger

Coecient Matrices and Aug-mented Matrices

Thexiactually carry no information, the system is

completely described by theaijandbi,i= 1;:::m, j= 1;:::;n.

We thus use thematrix of coecients, wich is

anmnarray containing the coecients of the equations. 0 B BB@a

11a12::: a1n

a21a22::: a2n............ a m1am2::: amn1 C CCA

We also have theAugmented Matrix, which in-

cludes thebion the right: 0 B BB@a

11a12::: a1nb1

a21a22::: a2nb2............... a m1am2::: amnbm1 C CCA

The augmented matrix contains all the informa-

tion necessary to solve the system. 3

1.2 Gaussian Elimination P. Danziger

1. Find the matrix of coecients and the aug-

mented matrix for the following system. x+ 2y3z= 1 +y+z= 1 x+y+z= 0

This system of equations has coecient ma-

trix:0 B @1 23 0 1 1

1 1 11

C A and Augmented matrix: 0 B @1 231

0 1 11

1 1 101

C A

2. Find the augmented matrix for the following

system. x+2z= 1 +yz= 0

This system of equations has Augmented ma-

trix: 1 021

0 110!

4

1.2 Gaussian Elimination P. Danziger

3. Given the following augmented matrix nd the

original system of equations. 0 B @1 23 0 11 1 101 C A

The system is

x+ 2y=3 y= 1 x+y= 0

This is a system of 3 equations in 2 unknowns.

It is inconsistent (no solution), since by the second equationy= 1, the third equation then tells us thatx=1, but then the rst equation states (substituting inx=1 andy= 1):1 + 2 = 3, which is not true. 5

1.2 Gaussian Elimination P. Danziger

Note that each ow of the augmented matrix cor-

responds to one of the original equations.

Each column contains the all the coecients of

a given variable in the system. We say that this columncorrespondsto this variable.

Example 2

x+ 2y=3 y= 1 x+y= 00 B @1 23 0 11 1 101 C A

The rst row corresponds tox, the second cor-

responds toyand the third corresponds to the constants. 6

1.2 Gaussian Elimination P. Danziger

Elementary Row Operations

There are three basic operations we can preform

on equations, these correspond toRow Operations on the corresponding matrices.

1. We can multiply an equation by a constant

Multiply a row by a constant.

2. Add a multiple of one equation to another

replace a row by itself plus a multiple of an- other row.

3. Interchange the order of equationsInter-

change two rows.

NotationWe generally denote theithrow of the

matrix byRi. Letcbe a constant, and 1i;jm then R i!Ri+cRjmeans replace Rowiby rowiplus ctimes rowj. R i!cRimeans replace rowiwithctimes rowi. R i$Rjmeans interchange rowiwith rowj. 7

1.2 Gaussian Elimination P. Danziger

Note that preforming any of these operations does

not change the solution to the original system of equations.

When using row operations always indicate

the operation you have used!

Example 3

1. 0 B @1 1 33

2 2 33

1 1 111

C

AR2!R22R1

R3!R3R1!0

B @1 1 33 0 033

0 0221

C A 2. 0 B @1 1 33

2 2 33

1 1 111

C

AR1$R2!0

B @2 2 33

1 1 33

1 1 111

C A 3. 0 B @1 1 33

2 2 33

1 1 111

C

AR2!2R2!0

B @2 2 33

4 4 66

1 1 111

C A

Never operate on the same row twice in one step.

8

1.2 Gaussian Elimination P. Danziger

Row Echelon Form

Denition 41. A matrix is in Row Echelon Form(REF) if all of the following hold: (a) Any rows consisting entirely of 0's appear at the bottom. (b) In any non-zero row the rst number, from the left, is a one. Called theleading oneor pivot.quotesdbs_dbs47.pdfusesText_47

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