Simultaneous Linear Equations
description
Transcript of Simultaneous Linear Equations
Simultaneous Linear Equations
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The name of the person in the picture is
A. B. C. D. E.
20% 20% 20%20%20%
A. A$AP Rocky
B. Kid Cudi
C. MC Hammer
D. T.I.
E. Vanilla Ice
The size of matrix
43
1. 2. 3. 4.
25% 25%25%25%
34
33
44
A.
B.
C.
D.
8765
4329
8764
is
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The c32 entity of the matrix
1. 2. 3. 4.
25% 25%25%25%
9.82.73.65
4329
87611.4
][C
A. 2B. 3
C. 6.3
D. does not exist
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Given
1. 2. 3.
33% 33%33%
395
263][A
62.98
362][B
then if [C]=[A]+[B], c12=
1. 0
2. 6
3. 12
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A square matrix [A] is lower triangular if
jiaij ,0
1. 2. 3. 4.
25% 25%25%25%
ijaij ,0
jiaij ,0ijaij ,0
A.
B.
C.
D.
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An identity matrix [I] needs to satisfy the following
jiI ij ,0
1. 2. 3. 4.
25% 25%25%25%
jiI ij ,1
A.
B.
C.
D. all of the above
matrix is square
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Given
1. 2. 3. 4.
25% 25%25%25%
then if [C]=[A][B], then c31= .A. -57
B. -45
C. 57
D. Does not exist
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54
79
34
,
956
821
364
BA
The following system of equations x + y=26x + 6y=12has solution(s).
1. 2. 3. 4.
25% 25%25%25%1. no2. one3. more than one but finite number
of4. infinite
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PHYSICAL PROBLEMS
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Truss Problem
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Pressure vessel problem
a
b
a
b
c
0
007.0
0
10887.7
106057.3102857.400
15384.05.615384.05.6
104619.5102857.4104619.5102857.4
00102307.9102857.4 3
4
3
2
1
57
5757
57
c
c
c
c
r
crcu 2
11
r
crcu 4
32
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Polynomial Regression
2210 TaTaaα
We are to fit the data to the polynomial regression model
n
iii
n
iii
n
ii
n
ii
n
ii
n
ii
n
ii
n
ii
n
ii
n
ii
n
ii
T
T
a
a
a
TTT
TTT
TTn
1
2
1
1
2
1
0
1
4
1
3
1
2
1
3
1
2
1
1
2
1
)()(),()( 112211 nnn-n- ,αT,,αT...,,αT,,αT
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END
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Simultaneous Linear Equations
Gaussian Elimination(Naïve and the Not That So Innocent Also)
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The goal of forward elimination steps in Naïve Gauss elimination method is to reduce the coefficient matrix to a (an) _________ matrix.
1. 2. 3. 4.
25% 25%25%25%
1. diagonal
2. identity
3. lower triangular
4. upper triangular
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One of the pitfalls of Naïve Gauss Elimination method is
0% 0%0%
1. large truncation error
2. large round-off error
3. not able to solve equations with a noninvertible coefficient matrix
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Increasing the precision of numbers from single to double in the Naïve Gaussian elimination method
1 2 3
33% 33%33%
1. avoids division by zero
2. decreases round-off error
3. allows equations with a noninvertible coefficient matrix to be solved
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Division by zero during forward elimination steps in Naïve Gaussian elimination for [A][X]=[C] implies the coefficient matrix [A]
1. 2. 3.
33% 33%33%1. is invertible
2. is not invertible
3. cannot be determined to be invertible or not
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Division by zero during forward elimination steps in Gaussian elimination with partial pivoting of the set of equations [A][X]=[C] implies the coefficient matrix [A]
1. 2. 3.
33% 33%33%1. is invertible
2. is not invertible
3. cannot be determined to be invertible or not
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Using 3 significant digit with chopping at all stages, the result for the following calculation is
A. B. C. D.
25% 25%25%25%8
99.1456.3095.61
x
A. -0.0988
B. -0.0978
C. -0.0969
D. -0.0962
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Using 3 significant digits with rounding-off at all stages, the result for the following calculation is
A. B. C. D.
25% 25%25%25%
8
99.1456.3095.61
x
A. -0.0988
B. -0.0978
C. -0.0969
D. -0.0962
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Determinants If a multiple of one row of [A]nxn is added or subtracted to another row of [A]nxn to result in [B]nxn then det(A)=det(B)
The determinant of an upper triangular matrix [A]nxn is given by nnii aaaa ......Adet 2211
n
iiia
1
Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.
nnnn UA
UA detdet http://nm.mathforcollege.com
Simultaneous Linear Equations
LU Decomposition
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You thought you have parking problems. Frank Ocean is scared to park when __________ is around.
A. B. C. D.
25% 25%25%25%
A. A$AP Rocky
B. Adele
C. Chris Brown
D. Hillary Clinton
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Truss Problem
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If you have n equations and n unknowns, the computation time for forward substitution is approximately proportional to
A. B. C.
33% 33%33%A. 4n
B. 4n2
C. 4n3
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If you have a nxn matrix, the computation time for decomposing the matrix to LU is approximately proportional to
A. B. C.
33% 33%33%A. 8n/3
B. 8n2/3
C. 8n3/3
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LU decomposition method is computationally more efficient than Naïve Gauss elimination for solving
1. 2. 3.
33% 33%33%
A. a single set of simultaneous linear equations
B. multiple sets of simultaneous linear equations with different coefficient matrices and same right hand side vectors.
C. multiple sets of simultaneous linear equations with same coefficient matrix and different right hand side vectors
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For a given 1700 x 1700 matrix [A], assume that it takes about 16 seconds to find the inverse of [A] by the use of the [L][U] decomposition method. Now you try to use the Gaussian Elimination method to accomplish the same task. It will now take approximately ____ seconds.
1 2 3 4
25% 25%25%25%
A. 4
B. 64
C. 6800
D. 27200
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For a given 1700 x 1700 matrix [A], assume that it takes about 16 seconds to find the inverse of [A] by the use of the [L][U] decomposition method. The approximate time in seconds that all the forward substitutions take out of the 16 seconds is
1 2 3 4
25% 25%25%25%A. 4
B. 6
C. 8
D. 12
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THE END
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Consider there are only two computer companies in a country. The companies are named Dude and Imac. Each year, company Dude keeps 1/5th of its customers, while the rest switch to Imac. Each year, Imac keeps 1/3rd of its customers, while the rest switch to Dude. If in 2003, Dude had 1/6th of the market and Imac had 5/6th of the marker, what will be share of Dude computers when the market becomes stable?
1. 2. 3. 4.
25% 25%25%25%
1. 37/902. 5/113. 6/114. 53/90
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You know Lady Gaga; Who is Shady Gaga
A. B. C. D.
25% 25%25%25%
A. Lady Gaga’s sister
B. A person who looks bad with their sunglasses on
C. A person who looks good with sunglasses but bad once he/she takes the sunglasses off
D. That is what Alejandro calls Lady Gaga
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Given
1. 2. 3. 4.
25% 25%25%25%
01.100
010
001
A
then [A] is a matrix.
A. diagonal
B. identity
C. lower triangular
D. upper triangular
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A square matrix nnA ][
1. 2. 3. 4. 5.
20% 20% 20%20%20%
is diagonally dominant if
niaan
jij
ijii ,.....,2,1,1
n
jij
ijii
n
jij
ijii
nianyforaa
andniaa
1
1
,....,2,1 ,
,.....,2,1,
n
jijii
n
jijii
nianyforaa
andniaa
1
1
,....,2,1 ,
,.....,2,1,
niaan
jijii ,.....,2,1,
1
1.
2.
3.
4.
5.
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The following data is given for the velocity of the rocket as a function of time. To find the velocity at t=21s, you are asked to use a quadratic polynomial v(t)=at2+bt+c to approximate the velocity profile.
35.517
78.362
04.227
120400
115225
114176
c
b
a
1. 2. 3. 4.
25% 25%25%25%
97.602
35.517
78.362
130900
120400
115225
c
b
a
35.517
78.362
0
120400
115225
100
c
b
a
67.901
97.602
35.517
1351225
130900
120400
c
b
a
t (s) 0 14 15 20 30 35
v m/s 0 227.04
362.78
517.35
602.97
901.67
A.
B.
C.
D.
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An example of upper triangular matrix is
300
600
532
1. 2. 3. 4.
25% 25%25%25%
320
600
532
320
326
532
A.
B.
C.
D. none of the above
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An example of lower triangular matrix is
654
003
002
1. 2. 3. 4.
25% 25%25%25%
654
023
092
009
063
052
A.
B.
C.
D. none of the above
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Three kids-Jim, Corey and David receive an inheritance of $2,253,453. The money is put in three trusts but is not divided equally to begin with. Corey’s trust is three times that of David’s because Corey made and A in Dr.Kaw’s class. Each trust is put in and interest generating investment. The total interest of all the three trusts combined at the end of the first year is $190,740.57 . The equations to find the trust money of Jim (J), Corey (C) and David (D) in matrix form is
57.740,190
0
453,253,2
011.008.006.0
310
111
D
C
J
1. 2. 3. 4.
25% 25%25%25%
57.740,190
0
453,253,2
011.008.006.0
130
111
D
C
J
57.740,190
0
453,253,2
1186
310
111
D
C
J
57.740,190
0
453,253,2
1186
130
111
D
C
J
A.
B.
C.
D.
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Is how much you are loaded up related to test score?
Test Score vs Hours of Effort Expected
y = -0.0402x + 72.843
R2 = 0.002720
40
60
80
100
0 20 40 60 80 100
Hours of Effort Expected
Tes
t S
core
1
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Final Grade vs Test#1 GradeFinal Grade vs Test#1 Grade
y = 0.5574x + 36.662
R2 = 0.2783
40
60
80
100
40 50 60 70 80 90 100
Test#1 Grade
Fin
al G
rade
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Determinants If a multiple of one row of [A]nxn is added or subtracted to another row of [A]nxn to result in [B]nxn then det(A)=det(B)
The determinant of an upper triangular matrix [A]nxn is given by nnii aaaa ......Adet 2211
n
iiia
1
Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.
nnnn UA
UA detdet http://nm.mathforcollege.com
The name of the person in the picture is
A. B. C. D.
25% 25%25%25%
A. Yung Joc
B. Kid Cudi
C. T.I.
D. MC Hammer
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Kanye West is a genius except
A. B. C. D.
25% 25%25%25%
A. He grabbed Taylor Swift’s mike at the VMAs
B. Has diamonds drilled to his bottom teeth
C. Sings about Mama’s boyfriend
D. All of the above
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Example of a PoemBoom Boom Pow, That is how I feel when I come to class, Glad that I have a lot of mass. I need to integrate my work and life, Differentiate between love and strife, Interpolate when my friend whines, Isn’t that same as reading between the lines?
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This Kiss – Faith Hill
It's a feeling like thisIt's centrifugal motionIt's perpetual blissIt's that pivotal momentIt's, ah unthinkableThis kiss, this kissUnsinkableThis kiss, this kiss
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Given
1. 2. 3.
33% 33%33%
395
263][A
62.98
362][B
then if [C]=[A]-[B], c23=
A. -3
B. 3
C. 9
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A square matrix [A] is upper triangular if
jiaij ,0
1. 2. 3. 4.
25% 25%25%25%
ijaij ,0
jiaij ,0ijaij ,0
1.
2.
3.
4.
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