Consider a set of algebraic equations Mx = b where 0.778 -1.222 -0.889 2.111 1 M= 0.889 -0.111 The singular value decomposition of M is M = USV where [-0.540 U=-0.260 -0.774 0.577 0.800 0.613 0.577 0.161 0.577 [2.91 %3= 1.30 0 -0.461 0.348 -0.816 V = 0.034 -0.912 -0.408 0.887 0.216 -0.408 ) Based on this decomposition, show that if an exact solution to Mx = b exists, then it will not be unique. (ii) Determine any value for the vector b such that a solution to Mx = b will exist? %3D (iii) What is the set of all b such that a solution to Mx = b will exist? (iv) For the b of part (ii), determine a solution to Mx = b.
Consider a set of algebraic equations Mx = b where 0.778 -1.222 -0.889 2.111 1 M= 0.889 -0.111 The singular value decomposition of M is M = USV where [-0.540 U=-0.260 -0.774 0.577 0.800 0.613 0.577 0.161 0.577 [2.91 %3= 1.30 0 -0.461 0.348 -0.816 V = 0.034 -0.912 -0.408 0.887 0.216 -0.408 ) Based on this decomposition, show that if an exact solution to Mx = b exists, then it will not be unique. (ii) Determine any value for the vector b such that a solution to Mx = b will exist? %3D (iii) What is the set of all b such that a solution to Mx = b will exist? (iv) For the b of part (ii), determine a solution to Mx = b.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you please solve part c&d I include one solution because I can just add two images no more, so sorry for it. I have a solution thank you
![3.17. Consider a set of algebraic equations Mx =b where
-0.778 -1.222
-0.889
2.111
M=
0.889
-0.111
The singular value decomposition of M is M =USV* where
T-0.540
U=-0.260 -0.774 0.577
0.800
0.613
0.577
0.161
0.577
[2.91
%=
1.30 0
-0.461
0.348
-0.816
-0.912 -0.408
-0.408
V =
0.034
0.887
0.216
(1) Based on this decomposition, show that if an exact solution to Mx = b
exists, then it will not be unique.
(ii) Determine any value for the vector b such that a solution to Mx = b will
exist?
(iii) What is the set of all b such that a solution to Mx=b will exist?
(iv) For the b of part (ii), determine a solution to Mx= b.
(v) Using the b of part (ii), determine all solutions to Mx = b. Rather than per-
form the calculations, based on the above numbers, please use the notation
below to develop a formula for the solution.
07
U= [u, uz uz]
S =
V = [v, vz vj]
02
where u; and v, are the columns of U and V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65a54057-de91-4407-81dc-84071bb6029d%2F2e180e0a-459b-429d-aec1-5ec8b2a7b033%2F82dxtt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.17. Consider a set of algebraic equations Mx =b where
-0.778 -1.222
-0.889
2.111
M=
0.889
-0.111
The singular value decomposition of M is M =USV* where
T-0.540
U=-0.260 -0.774 0.577
0.800
0.613
0.577
0.161
0.577
[2.91
%=
1.30 0
-0.461
0.348
-0.816
-0.912 -0.408
-0.408
V =
0.034
0.887
0.216
(1) Based on this decomposition, show that if an exact solution to Mx = b
exists, then it will not be unique.
(ii) Determine any value for the vector b such that a solution to Mx = b will
exist?
(iii) What is the set of all b such that a solution to Mx=b will exist?
(iv) For the b of part (ii), determine a solution to Mx= b.
(v) Using the b of part (ii), determine all solutions to Mx = b. Rather than per-
form the calculations, based on the above numbers, please use the notation
below to develop a formula for the solution.
07
U= [u, uz uz]
S =
V = [v, vz vj]
02
where u; and v, are the columns of U and V.
Transcribed Image Text:ull Verizon ?
9:12 AM
9 92%
Step2
b)
And Mx = b=>USV"x=b
=>U"(USv"x)=U*b
=>S(v*x)=U*b (as UTU=I)
Let z=V"x and d=U™b.
=>Sz=d
So, discussing the non-uniqueness property of the
solution of Mx=b is the same as discussing
for Sz=d.
Let b = 62
b3
-0. 540b1 – 0. 260b, + 0. 800b3
0. 613b1
Then d =
0. 774b2 +
161b3
0.577b1 + 0. 577b, + 0. 577b3
?
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