maths II sutd lecture 2

8/10/2019 maths II sutd lecture 2

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Singapore University of

Technology & Design

MATH 10.004Elimination & Augmented Form

Cohort 2

Meyer, Sections 1.2-1.3


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Elimination Matrices Elimination with Matrices Summary

LEARNING OBJECTIVES

After this cohort you will be able to ...

solve system of linear equations using elimination and back

substitution.

express systems in terms of matrices and perform basic

operations with matrices.

solve system of linear equations using augmented matrices.

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ELIMINATION

Purpose is to provide a systematic way to solve systems of linearequations.

Commonly credited to Carl Friedrich Gauss, but first appearance

was in The Nine Chapters on the Mathematical Arts, an early

Chinese mathematics book composed by several authors and

completed around the 1st century.

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ELIMINATION

Our aim is to simplify equations by performing the following types

ofrow operations:

(I) Exchange two equations.

(II) Multiply an equation by a non-zero constant.

(III) Add a multiple of one equation to another equation.

In particular, we want to eliminate variables to allow for easier

subsitution.

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EXAMPLE

Consider the following linear system

2x1` x2` x3 54x16x2 2

2x1`7x2`2x3 9

Well solve it via a series of elimination steps on the next slide.

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Original system:

2x1` x2` x3 5 (1)4x16x2 2 (2)

2x1`7x2`2x3 9 (3)

We proceed byforward

elimination:

(2)-2(1)(2):

2x1` x2` x3 5

8x22x3 12

2x1`7x2`2x3 9

(3)+(1)(3):

2x1` x2` x3 5

8x22x3 12

8x2`3x3 14

(3)+(2)(3):

2x1` x2` x3 5

8x22x3 12

1x3 2

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Eli i i M i Eli i i i h M i S


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We now have the upper triangular system

2x1` x2` x3 5

8x2 2x3 12

1x3 2

which we may then solve viaback substitution:

1. From the third equation we havex32.

2. Pluggingx32 into the second equation, we then have

x2 14x3` 32 12` 32 1.

3. Plugging bothx32 andx21 into the first equation gives usx1

12

px2`x3q ` 52

1.

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Eli i ti M t i Eli i ti ith M t i S


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GAUSS-JORDAN METHOD

TheGauss-Jordan methodsimplifies back substitution.

2x1` x2` x3 5

8x2 2x3 12

1x3 2

2x1` x2 3

x2 1

x32

2x1` x2` 3

8x2 8

x3 2

x1 1

x2 1

x32

The leading coefficient in each equation is a 1.

Every coefficient of the same unknown above these leading

terms is zero.8



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ACTIVITY 1: DOCKING A SPACE POD (15 MINUTES)

You find yourself in deep space piloting a small space pod that you

would like to dock to the mother ship.

You are currently stationary and your navigation tools define an

px, y, zqcoordinate system relative to your current position. Thedocking location is 4, 10, and 17 meters away (in x,y, andz).

You have 3 thrusters at your control. For each second you fire each

thruster the pod will move, in thex,y, andzdirections:

Thruster A: 1, 2, and 3 meters,

Thruster B: 1, 3, and 6 meters, Thruster C: 2, 6, and 10 meters.

Assuming a simple additive model of the interaction of the thrusters,

use elimination to find how many seconds each thruster needs to be

fired to move the pod to the dock.9



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ACTIVITY 1: DOCKING A SPACE POD

Lets1,s2, ands3 be the time that each of the thrusters is fired. Using

matrix notation, the problem to solve may be written as:

s1`s2`2s3 4

2s1`3s2`6s310

3s1`6s2`10s317or after elimination:

1. p2q 2p1q p2q

2. p3q 3p1q p3q

3. p3q 3p2q p3q

s1`s2`2s3 4

s2`

2s3

2

2s3 1

which through back substitution then gives us the solution s12,s21, ands31{2.

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MATRICES

Amatrixis a rectangular array of scalars.

If the matrix hasmrows andncolumns, we say that the size of

the matrix ismn.

The matrix is square ifmn.

The scalar in theith row andjth column is called thepi,jq-entry ofthe matrix.

Am n

a11 . . . a1j . . . a1n...

......

ai1 . . . aij . . . ain...

......

am1 . . . amj . . . amn

fiffiffiffiffiffiffifl

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SUM OF MATRICES

Matrices of the same dimensions can be added.

IfA

a11 . . . a1n

..

.

..

.am1 . . . amn

fiffifl

andB

b11 . . . b1n

..

.

..

.bm1 . . . bmn

fiffifl

,

then

A`Ba11`b11 . . . a1n` b1n

.

..

.

..am1`bm1 . . . amn` bmn

fiffifl .

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PRODUCT OF A MATRIX BY A SCALAR

Matrices can be multiplied by a scalar. If A

a11 . . . a1n...

...

am1 . . . amn

fiffifl

and P R, then

A

a11 . . . a1n..

.

..

.am1 . . . amn

fiffifl .

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PROPERTIES OF MATRIX OPERATIONS

Matrices obey the following laws related to addition:

A`BB`A (the commutative law)

cpA`Bq cA`cB(the distributive law)

A` pB`Cq pA`Bq `CA`B`C(the associative law)

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TRANSPOSE

Given a matrixA, itstransposeAT is obtained by interchanging the

role of rows and columns. For instance:

A24

1 2 7 03 2 1 6

AT

42

1 3

2 27 1

0 6

fiffiffifl .

pAT

qT

A pABqT AT BT

pAqT AT

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VECTORS AS MATRICES

Arow vectorof sizenis a 1nmatrix.

Acolumn vectorof sizemis anm1 matrix.

Ifvis a column vector, then vT is a row vector.

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PRODUCT OF MATRICES

Matrices can be multiplied if the number of columns of the first matrix

is equal to the number of rows of the second. If

Am n

a11 . . . a1n

......

am1 . . . amn

fi

ffifland B

n k

b11 . . . b1k

......

bn1 . . . bnk

fi

ffifl,then

Cm k

AB

is such that its genericpi,jq-entry has the form

cijn

h1

aihbhjAi B jApi, :q Bp:,jq.

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SIMPLE MATRIX MULTIPLICATION

For column vectorsu,v,wandy:

vTwv1 v2 v3

w1w2w3

fifl v1w1`v2w2`v3w3

uT

vT

w

u1 u2 u3

v1 v2 v3

w1w2w3

fifl

u1w1`u2w2`u3w3v1w1`v2w2`v3w3

vT

w y

v1 v2 v3

w1 y1w2 y2w3 y3

fifl

v1w1`v2w2`v3w3 v1y1`v2y2`v3y318



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SYSTEM OF LINEAR EQUATIONS

Alinear systemof mequations innunknowns is written as

a11x1à12x2` a1nxnb1

a21x1à22x2` a2nxnb2...

am1x1àm2x2` amnxnbm.

which is a mathematical way of expressingmlinear equality

constraints that thenvariablesxi, iP t1, . . . , nu, need to satisfy.

In matrix notation, we can write this compactly as

Axb,

whereA P Rm n,xP Rn, andbP Rm.19



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MATRIX FORM

We can rewrite the system:

2x1` x2` x3 5 (4)

4x16x2 2 (5)

2x1`7x2`2x3 9 (6)

as

Axb,

where

A

2 1 14 6 0

2 7 2

fifl , x

x1x2x3

fifl , b

52

9

fifl .

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ACTIVITY 2: MATRICES FOR DATA MANAGEMENT (20 MINUTES)

Create a matrixSthat records the day-end sales formstoresselling the samen items.

Price information will be stored in a n 1 matrix (vector) calledp.

What does theith

row ofStell you? And thejth

column?

What does thesijelement in the matrix tell you?

Find matrix operations to calculate:

A vector containing number of itemjs sold at each store. A vector containing the total revenue of each store.

The total revenue from all stores.

The total revenue of store i.

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ACTIVITY 2: MATRICES FOR DATA MANAGEMENT

S

s11 . . . s1n

......

sm1 . . . smn

fiffifl and p

p1

p2...

pn

fiffiffiffifl

Theith row gives the number of each item sold at store i.

Thejth column gives the number of item js sold at each of thestores.

The elementsijgives the number of itemjs sold at storei.

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A vector containing number of item js sold at each store:

Sej

s11 . . . s1j . . . s1n...

......

sm1 . . . smj . . . smn

fiffifl

0...

1..

.0

fiffiffiffiffiffiffifl

s1j...

smj

fiffifl

A vector containing the total revenue of each store:

Sp

s11 . . . s1n... ...sm1 . . . smn

fiffiflp1...pn

fiffifl rev1...

revm

fiffifl ,

where revkis the revenue made from all items at storek.

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The total revenue from all stores:

1TmSp

1 . . . 1

s11 . . . s1n...

...

sm1 . . . smn

fiffifl

p1...

pn

fiffifl

m

k1

revk.

The total revenue of storei:

eTiSp

0 . . . 1 . . . 0

s11 . . . s1n...

..

.si1 . . . sin

......

sm1 . . . smn

fiffiffiffiffiffiffiflp1...pn

fiffifl revi.

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ACTIVITY 2B (5 MINUTES)

Enter these commands into SCILAB, MATLAB, etc and discuss their

meaning:

clear all;

sales=[500 520 128 58; 850 600 54 32]

prices=[1.55 2.35 1.5 3.5]

revshop=sales*prices

revtot=[1 1]*revshop

revshopfruit=sales(:,1:3)*prices(1:3)revtotfruit=[1 1]*revshopfruit

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ACTIVITY 3(10 MINUTES)

Consider an electric circuit which takes input voltageVin and currentIinand produces output voltageVout and currentIout.

For a series circuit:

V2V1I1R1 andI2I1.

For a parallel circuit:

V3V2 andI3I2V2{R2.

Define a (transfer) matrixA such that:

V2

I2

A1

V1

I1

,

V3

I3

A2

V2

I2

,

V3

I3

A3

V1

I1

for the series circuit, parallel circuit, and combined circuit, respectively.

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ACTIVITY 3

According to Ohms law and Kirchhoffs circuit laws:

For the series circuit,V2V1I1R1,I2I1:

The transfer matrix of a series circuit is A1

1 R10 1

.

For the parallel circuit,V3V2,I3I2V2{R2:

The transfer matrix of a parallel circuit is A2

1 0

1{R2 1

.

We can use matrix multiplication to find the total transfer function:

V3

I3

A2

V2

I2

A2A1

loomoonA3V1

I1

, A3

1 R1

1{R2 1`R1{R2

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AUGMENTED FORM

Given the equationAxb

2 1 1

4 6 02 7 2

fi

fl

x1

x2

x3

fi

fl

5

29

fi

fl,

we callA b

theaugmented matrixof the system.

A b

2 1 1 54 6 0 2

2 7 2 9

fifl

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Original system:

2x1` x2` x3 5 (1)

4x16x2 2 (2)

2x1`7x2`2x3 9 (3)


elimination:

(2)-2(1)(2):

2x1` x2` x3 5

8x22x3 12

2x1`7x2`2x3 9

(3)+(1)(3):

2x1` x2` x3 58x22x3 12

8x2`3x3 14

(3)+(2)(3):

2x1` x2` x3 5

8x22x3 12

1x3 2

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Original system:

2 1 1 54 6 0 22 7 2 9

fifl


elimination:

(2)-2(1)(2):

2 1 1 50 8 2 12

2 7 2 9

fifl

(3)+(1)(3):

2 1 1 50 8 2 12

0 8 3 14

fifl

(3)+(2)(3):

2 1 1 5

0 8 2 120 0 1 2fifl

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ELIMINATION WITH MATRICES

Using matrix notation, we started with the original system:

Ax

2 1 14 6 0

2 7 2

fifl x

52

9

fifl b,

and through a series of elementary row operations transformed it to

theequivalent system:

Ux

2 1 10 8 2

0 0 1

fifl x

512

2

fifl c.

The Gauss-Jordan method on the augmented matrix works the sameway as before:

Rx

1 0 0

0 1 0

0 0 1

fi

flx

1

1

2

fi

fld.

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ACTIVITY 4: AUGMENTED MATRICES (15 MINUTES)

Solve Activity 1: Docking a Space Pod using augmented matrices.

Swap rows 2 3 and solve through elimination again.

Are the final triangular systems the same?

Are the solutions the same?

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ACTIVITY 4: AUGMENTED MATRICES

Lets1,s2, ands3 be the time that each of the thrusters is fired. Using

matrix notation, the problem to solve may be written as:1 1 2 42 3 6 10

3 6 10 17

fifl

1 1 2 40 1 2 2

0 0 2 1

fifl

or after switching rows 2 3:

1 1 2 4

3 6 10 17

2 3 6 10

fi

fl

1 1 2 4

0 3 4 5

0 0 23

13

fi

flElimination is not unique!

Either way substitution then gives us the same solution:

ps1, s2, s3q p2, 1, 1{2q.

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ACTIVITY 5: PRACTICE (10 MINUTES)

Using elimination on augmented matrices solve the following system

of linear equations.

x1`2x2` 8x37x4 2

3x1`2x2`12x35x4 6

x1` x2` x35x4 10

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ACTIVITY 5: PRACTICE

1 2 8 7 23 2 12 5 6

1 1 1 5 10

fifl

p2q 3p1q p3q ` p1q

1 2 8 7 20 4 12 16 120 3 9 12 12

fi

fl 3p2q 4p3q

1 2 8 7 20 12 36 48 36

0 12 36 48 48

fifl

p3q ` p2q

1 2 8 7 20 12 36 48 36

0 0 0 0 12

fifl

This system is inconsistent,

therefore it has no solution!

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SUMMARY

Elimination provides a systematic way to solve systems of linear

equations.

Systems of linear equations can be represented by a matrix

equation.

Basic definitions of matrices and their properties.

Augmented form is a compact way to solve linear systems withelimination.

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maths II sutd lecture 2