Autar Kaw Humberto Isaza - MATH FOR COLLEGEmathforcollege.com › nm › mws › gen › 04sle ›...
Transcript of Autar Kaw Humberto Isaza - MATH FOR COLLEGEmathforcollege.com › nm › mws › gen › 04sle ›...
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Autar KawHumberto Isaza
http://nm.MathForCollege.comTransforming Numerical Methods Education for STEM Undergraduates
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http://nm.MathForCollege.com
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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.
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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
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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
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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, ==
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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
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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
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Add the two vectors
and
=
1432
A
−
=
732
5
B
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Solution
−
+
=+
732
5
1432
BA
++−+
=
71342352
=
8717
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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
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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
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A null vector is where all the components of the vector are zero.
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Give an example of a null vector or zero vector.
Solution
The vector
Is an example of a zero or null vector
0000
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A unit vector is defined as
where
U
=
nu
uu
U
2
1
122322
21 =++++ nuuuu
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Give examples of 3-dimensional unit column vectors.
Solution
Examples include
etc. ,010
,
02
12
1
,001
,
313
13
1
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If k is a scalar and is a n-dimensional vector, then
=
na
aa
kAk
2
1
=
nka
kaka
2
1
A
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What is ifA
2
=
52025
A
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Solution
=
52025
22A
×××
=52202252
=
104050
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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
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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
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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
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Find the linear combinationsa) andb)where
BA
−
CBA
3−+
=
=
=
21
10,
211
,632
CBA
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Solution
a)
−
=−
211
632
BA
−−−
=261312
=
421
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Solution
b)
−
+
=−+
21
103
211
632
3CBA
−+−+−+
=6263133012
−=
2127
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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
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Are the three vectors
linearly independent?
=
=
=
111
,1285
,1446425
321 AAA
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Solution
Writing the linear combination of the three vectors
gives
=
+
+
000
111
1285
1446425
321 kkk
=
++++++
000
12144864525
321
321
321
kkkkkkkkk
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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 =
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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
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Are the three vectors
linearly independent?
=
=
=
24146
,752
,521
321 AAA
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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
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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)
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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 −=
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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
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(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
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Are the three vectors
linearly dependent?
=
=
=
211
,1385
,896425
321 AAA
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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
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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
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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.
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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
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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
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What is the rank of
Solution
From inspection
that implies
=
=
=
533
,422
,211
321 AAA
12 2AA
=→
.002 321
=+− AAA
,
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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
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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
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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
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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
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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
,,, 21mkk ,,1
02211
=+++ mm AkAkAk
0≠pk miki ,,1, = pi =
.11
11
22
mp
mp
p
pp
p
p
pp Ak
kA
kk
Ak
kA
kkA
−−−−−−= ++
−−
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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
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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
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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 ,.....,, 21CAxAx nn
=++ ..........11
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Write
as a linear combination of vectors.
8.106525 321 =++ xxx
2.177864 321 =++ xxx
2.27912144 321 =++ xxx
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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
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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
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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
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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 ($)
ABC
200250310
20.2330.5629.12
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Solution
The weight vector is given by
and the cost vector is given by
The total cost of the rubber would be the dot product of and C
W
( )310,250,200=W
( )12.29,56.30,23.20=C
( ) ( )12.29,56.30,23.20310,250,200 ⋅=⋅CW
)12.29)(310()56.30)(250()23.20)(200( ++=
2.902776404046 ++=
20.20713$=
.
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VectorAddition of vectorsRankDot ProductSubtraction of vectorsUnit vectorScalar multiplication of vectorsNull vectorLinear combination of vectorsLinearly independent vectors
VectorsUnary Matrix OperationsObjectivesWhat is a vector?Example 1When are two vectors equal?Example 2Example 2 (cont.) How do you add two vectors?Example 3Example 3 (cont.)Example 4Example 4 (cont.)What is a null vector?Example 5What is a unit vector?Example 6How do you multiply a vector by a scalar?Example 7Example 7 (cont.)Example 8Example 8 (cont.)What do you mean by a linear combination of vectors?Example 9Example 9 (cont.) Example 9 (cont.) What do you mean by vectors being linearly independent?Example 10Example 10 (cont.)Example 10 (cont.)Example 10 (cont.)Example 11Example 11 (cont.)Example 11 (cont.)Example 11 (cont.)Example 11 (cont.)Example 11 (cont.)Example 12Example 12 (cont.)Example 12 (cont.)What do you mean by the rank of a set of vectors?Example 13Example 14Example 15Example 15 (cont.)Linearly DependentLinearly IndependentLinearly Independent (cont.)Linearly DependentLinearly DependentLinearly Dependent (cont.)Linearly Dependent (cont.)Example 16Example 16 (cont.)Example 16 (cont.)Example 17Example 18Example 18 (cont.)Key Terms: