Autar KawHumberto Isaza

http://nm.MathForCollege.comTransforming Numerical Methods Education for STEM Undergraduates

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After reading this chapter, you should be able to:1. define a vector,2. add and subtract vectors,3. find linear combinations of vectors and their relationship to a set of equations,4. explain what it means to have a linearly independent set of vectors, and5. find the rank of a set of vectors.

A vector is a collection of numbers in a definite order. If it is a collection of nnumbers it is called a n-dimensional vector. So the vector given by

Is a n-dimensional column vector with n components, . The above is acolumn vector. A row vector [B] is of the form where is a n-dimensional row vector with n components

naaa ,......,, 21

A

=

na

aa

A

2

1

],....,,[ 21 nbbbB =

B

nbbb ,....,, 21

Give an example of a 3-dimensional column vector.

SolutionAssume a point in space is given by its (x,y,z) coordinates. Then if the value of

the column vector corresponding to the location of the points is5,2,3 === zyx

=

523

zyx

Two vectors and are equal if they are of the same dimension and if their corresponding components are equal.Given

and

Then if

A

B

=

na

aa

A

2

1

=

nb

bb

B

2

1

BA

= niba ii ,......,2,1, ==

What are the values of the unknown components in if

and

and

B

=

1432

A

=

4

1

43

b

b

B

BA

=

Solution

1 ,2 41 == bb

Two vectors can be added only if they are of the same dimension and the addition is given by

+

=+

nn b

bb

a

aa

BA

2

1

2

1

][][

+

++

=

nn ba

baba

22

11

Add the two vectors

and

=

1432

A

−

=

732

5

B

Solution

−

+

=+

732

5

1432

BA

++−+

=

71342352

=

8717

A store sells three brands of tires: Tirestone, Michigan and Copper. In quarter 1, the sales are given by the column vector

where the rows represent the three brands of tires sold – Tirestone, Michigan and Copper respectively. In quarter 2, the sales are given by

What is the total sale of each brand of tire in the first half of the year?

=

6525

1A

=

61020

2A

SolutionThe total sales would be given by

So the number of Tirestone tires sold is 45, Michigan is 15 and Copper is 12 in the first half of the year.

21 AAC

+=

+

=

61020

6525

+++

=66

1052025

=

121545

A null vector is where all the components of the vector are zero.

Give an example of a null vector or zero vector.

Solution

The vector

Is an example of a zero or null vector

0000

A unit vector is defined as

where

U

=

nu

uu

U

2

1

1223

22

21 =++++ nuuuu

Give examples of 3-dimensional unit column vectors.

Solution

Examples include

etc. ,010

,

02

12

1

,001

,

313

13

1

If k is a scalar and is a n-dimensional vector, then

=

na

aa

kAk

2

1

=

nka

kaka

2

1

A

What is ifA

2

=

52025

A

Solution

=

52025

22A

×××

=52202252

=

104050

A store sells three brands of tires: Tirestone, Michigan and Copper. In quarter 1, the sales are given by the column vector

If the goal is to increase the sales of all tires by at least 25% in the next quarter, how many of each brand should be sold?

=

62525

A

SolutionSince the goal is to increase the sales by 25%, one would multiply the vector by 1.25,

Since the number of tires must be an integer, we can say that the goal of sales is

A

=

62525

25.1B

=

5.725.3125.31

=

83232

B

Given

as m vectors of same dimension n, and if k1, k2,…, km are scalars, then

is a linear combination of the m vectors.

mAAA

,......,, 21

mm AkAkAk

+++ .......2211

Find the linear combinationsa) andb)where

BA

−

CBA

3−+

=

=

=

21

10,

211

,632

CBA

Solution

a)

−

=−

211

632

BA

−−−

=261312

=

421

Solution

b)

−

+

=−+

21

103

211

632

3CBA

−+−+−+

=6263133012

−=

2127

A set of vectors are considered to be linearly independent if

has only one solution of

mAAA

,,, 21

0.......2211

=+++ mm AkAkAk

0......21 ==== mkkk

Are the three vectors

linearly independent?

=

=

=

111

,1285

,1446425

321 AAA

Solution

Writing the linear combination of the three vectors

gives

=

+

+

000

111

1285

1446425

321 kkk

=

++++++

000

12144864525

321

321

321

kkkkkkkkk

The previous equations have only one solution, . However, how do we show that this is the only solution? This is shown below.The previous equations are

(1)(2)(3)

Subtracting Eqn (1) from Eqn (2) gives

(4)

Multiplying Eqn (1) by 8 and subtracting it from Eqn (2) that is first multiplied by 5 gives

(5)

0321 === kkk

0525 321 =++ kkk0864 321 =++ kkk

012144 321 =++ kkk

0339 21 =+ kk12 13kk −=

03120 31 =− kk

13 40kk =

Remember we found Eqn (4) and Eqn (5) just from Eqns (1) and (2).Substitution of Eqns (4) and (5) in Eqn (3) for and gives

This means that has to be zero, and coupled with (4) and (5), and are also zero. So the only solution is . The three vectors hence are linearly independent

040)13(12144 111 =+−+ kkk028 1 =k

01 =k

1k 2k 3k0321 === kkk

Are the three vectors

linearly independent?

=

=

=

24146

,752

,521

321 AAA

By inspection,

Or

So the linear combination

Has a non-zero solution

213 22 AAA

+=

022 321

=+−− AAA

0332211

=++ AkAkAk

1,2,2 321 =−=−= kkk

Hence, the set of vectors is linearly dependent.What if I cannot prove by inspection, what do I do? Put the linear combination of three vectors equal to the zero vector,

to give(1)(2)(3)

=

+

+

000

24146

752

521

321 kkk

062 321 =++ kkk01452 321 =++ kkk02475 321 =++ kkk

(1)(2)

Multiplying Eqn (1) by 2 and subtracting from Eqn (2) gives

(4)

Multiplying Eqn (1) by 2.5 and subtracting from Eqn (2) gives

(5)

(1)(2)

02 32 =+ kk

32 2kk −=

05.0 31 =−− kk

31 2kk −=

Remember we found Eqn (4) and Eqn (5) just from Eqns (1) and (2).Substitute Eqn (4) and (5) in Eqn (3) for and gives

This means any values satisfying Eqns (4) and (5) will satisfy Eqns (1), (2) and (3) simultaneously.For example, chose

thenfrom Eqn(4), andfrom Eqn(5).

(1)(2)

1k 2k

( ) ( ) 0242725 333 =+−+− kkk

0241410 333 =+−− kkk

00 =

63 =k122 −=k121 −=k

(1)(2)

Hence we have a nontrivial solution of . This implies the three given vectors are linearly dependent. Can you find another nontrivial solution?

What about the following three vectors?

Are they linearly dependent or linearly independent? Note that the only difference between this set of vectors and the previous one is thethird entry in the third vector. Hence, equations (4) and (5) are still valid. Whatconclusion do you draw when you plug in equations (4) and (5) in the thirdequation: ?What has changed?

[ ] [ ]61212321 −−=kkk

25146

,752

,521

02575 321 =++ kkk

Are the three vectors

linearly dependent?

=

=

=

211

,1385

,896425

321 AAA

Solution

Writing the linear combination of the three vectors and equating to zero vector

gives

=

+

+

000

211

1385

896425

321 kkk

=

++++++

000

21389864525

321

321

321

kkkkkkkkk

In addition to , one can find other solutions for which are not equal to zero. For example is also a solution. This implies

So the linear combination that gives us a zero vector consists of non-zero constants. Hence are linearly dependent

0321 === kkk 321 ,, kkk40,13,1 321 =−== kkk

=

+

−

000

211

401385

13896425

1

321 ,, AAA

From a set of n-dimensional vectors, the maximum number of linearly independentvectors in the set is called the rank of the set of vectors. Note that the rank of thevectors can never be greater than the vectors dimension.

What is the rank of

Solution

Since we found in Example 2.10 that are linearly dependent, the rank of the set of vectors is 3

=

=

=

111

,1285

,1446425

321 AAA

321 ,, AAA

321 ,, AAA

What is the rank of

Solution

In Example 2.12, we found that are linearly dependent, the rank of is hence not 3, and is less than 3. Is it 2? Let us choose

Linear combination of and equal to zero has only one solution. Therefore, the rank is 2.

321 ,, AAA

321 ,, AAA

=

=

1385

,896425

21 AA

1A

2A

=

=

=

211

,1385

,896425

321 AAA

What is the rank of

Solution

From inspection

that implies

=

=

=

533

,422

,211

321 AAA

12 2AA

=→

.002 321

=+− AAA

,

Hence

has a nontrivial solutionSo are linearly dependent, and hence the rank of the three vectors is not 3. Since

are linearly dependent, but

has trivial solution as the only solution. So are linearly independent. The rank of the above three vectors is 2.

,

.0332211

=++ AkAkAk

321 ,, AAA

12 2AA

=

21 and AA

.03311

=+ AkAk

31 and AA

Prove that if a set of vectors contains the null vector, the set of vectors is linearly dependent.

Let be a set of n-dimensional vectors, then

is a linear combination of the m vectors. Then assuming if is the zero or null vector, any value of coupled with will satisfy the above equation. Hence, the set of vectors is linearly dependent as more than one solution exists.

,

mAAA

,,........., 21

02211

=+++ mm AkAkAk

1A

1k 032 ==== mkkk

Prove that if a set of vectors are linearly independent, then a subset of the mvectors also has to be linearly independent.

Let this subset be

where Then if this subset is linearly dependent, the linear combination

Has a non-trivial solution.

,

apaa AAA

,,, 21

mp <

02211

=+++ appaa AkAkAk

So

also has a non-trivial solution too, where are the rest of the vectors.However, this is a contradiction. Therefore, a subset of linearly independent vectorscannot be linearly dependent.

,

( ) ampa AA

,,1+ )( pm −

00.......0 )1(2211

=++++++ + ampaappaa AAAkAkAk

Prove that if a set of vectors is linearly dependent, then at least one vector can be written as a linear combination of others.

Let be linearly dependent, then there exists a set of numbersnot all of which are zero for the linear combination

Let to give one of the non-zero values of , be for , then

and that proves the theorem.

,

mAAA

,,, 21

mkk ,,1

02211

=+++ mm AkAkAk

0≠pk miki ,,1, = pi =

.11

11

22

mp

mp

p

pp

p

p

pp A

kk

Ak

kA

kk

AkkA

−−−−−−= ++

−−

Prove that if the dimension of a set of vectors is less than the number of vectors in the set, then the set of vectors is linearly dependent.

Can you prove it?

How can vectors be used to write simultaneous linear equations?If a set of m linear equations with n unknowns is written as

,

11111 cxaxa nn =++

22121 cxaxa nn =++

nnmnm cxaxa =++11

Where are the unknowns, then in the vector notation they can be written as

where

,

nxxx ,,, 21

CAxAxAx nn

=+++ 2211

=

1

11

1

ma

aA

Where

The problem now becomes whether you can find the scalars such that the linear combination

,

=

1

11

1

ma

aA

=

2

12

2

ma

aA

=

mn

n

n

a

aA

1

=

mc

cC

1

1

nxxx ,.....,, 21

CAxAx nn

=++ ..........11

Write

as a linear combination of vectors.

8.106525 321 =++ xxx

2.177864 321 =++ xxx

2.27912144 321 =++ xxx

Solution

What is the definition of the dot product of two vectors?

=

++++++

2.2792.1778.106

12144864525

321

321

321

xxxxxxxxx

=

+

+

2.2792.1778.106

111

1285

1446425

321 xxx

Let and be two n-dimensional vectors. Then the dot product of the two vectors and is defined as

A dot product is also called an inner product or scalar.

[ ]naaaA ,,, 21

= [ ]nbbbB ,,, 21

=

A

B

∑=

=+++=⋅n

iiinn babababaBA

12211

Find the dot product of the two vectors = (4, 1, 2, 3) and = (3, 1, 7, 2).

Solution

= (4)(3)+(1)(1)+(2)(7)+(3)(2)= 33

A

B

( ) ( )2,7,1,33,2,1,4 ⋅=⋅ BA

A product line needs three types of rubber as given in the table below.

Use the definition of a dot product to find the total price of the rubber needed.

Rubber Type Weight (lbs) Cost per pound ($)

ABC

200250310

20.2330.5629.12

Solution

The weight vector is given by

and the cost vector is given by

The total cost of the rubber would be the dot product of and C

W

( )310,250,200=W

( )12.29,56.30,23.20=C

( ) ( )12.29,56.30,23.20310,250,200 ⋅=⋅CW

)12.29)(310()56.30)(250()23.20)(200( ++=

2.902776404046 ++=

20.20713$=

.

VectorAddition of vectorsRankDot ProductSubtraction of vectorsUnit vectorScalar multiplication of vectorsNull vectorLinear combination of vectorsLinearly independent vectors