Exercise 21.12. This exercise involves the overdetermined and underdetermined ideas when we combine linear systems. In each case below, if you don't think there is a reasonable guess for the behavior of the combined linear system (e.g. if it depends on the specific sizes of the underlying systems), you should explain why rather than giving an unreasonable guess. Accompany your answer with some informal reasoning (please don't write out anything involving long lists of variables or computations with them). (a) Suppose A,x = bị and A2y = b, are overdetermined linear systems. How many solutions do you A¡x = bị A2y = b2 %3D expect the combined system to have? (e.g. 0, 1, finitely many but maybe more than 1, or infinitely many) (b) Suppose A;x = bị is an overdetermined linear system and Aży = b2 is an underdetermined linear |A)x = bị , | A2y = b2 system. How many solutions do you expect the combined system to have? (Note that the systems of variables x and y are unrelated.) (c) Suppose A;x = bị and A,x = b, are linear systems of n equations in the same collection of n variables r1,..., Tn. How many solutions do you expect the combined system [A] |A2] to b2 have? (d) Suppose A¡x = bị and A2y = b2 are underdetermined linear systems where bị and bą are m-vectors for a common m. How many solutions do you expect the combined system [A A] = bị + b2 to have?
Exercise 21.12. This exercise involves the overdetermined and underdetermined ideas when we combine linear systems. In each case below, if you don't think there is a reasonable guess for the behavior of the combined linear system (e.g. if it depends on the specific sizes of the underlying systems), you should explain why rather than giving an unreasonable guess. Accompany your answer with some informal reasoning (please don't write out anything involving long lists of variables or computations with them). (a) Suppose A,x = bị and A2y = b, are overdetermined linear systems. How many solutions do you A¡x = bị A2y = b2 %3D expect the combined system to have? (e.g. 0, 1, finitely many but maybe more than 1, or infinitely many) (b) Suppose A;x = bị is an overdetermined linear system and Aży = b2 is an underdetermined linear |A)x = bị , | A2y = b2 system. How many solutions do you expect the combined system to have? (Note that the systems of variables x and y are unrelated.) (c) Suppose A;x = bị and A,x = b, are linear systems of n equations in the same collection of n variables r1,..., Tn. How many solutions do you expect the combined system [A] |A2] to b2 have? (d) Suppose A¡x = bị and A2y = b2 are underdetermined linear systems where bị and bą are m-vectors for a common m. How many solutions do you expect the combined system [A A] = bị + b2 to have?
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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![Exercise 21.12. This exercise involves the overdetermined and underdetermined ideas when we combine
linear systems. In each case below, if you don't think there is a reasonable guess for the behavior of the
combined linear system (e.g. if it depends on the specific sizes of the underlying systems), you should
explain why rather than giving an unreasonable guess. Accompany your answer with some informal
reasoning (please don't write out anything involving long lists of variables or computations with them).
(a) Suppose A,x = bị and A2y = b, are overdetermined linear systems. How many solutions do you
A¡x = bị
A2y = b2
%3D
expect the combined system
to have? (e.g. 0, 1, finitely many but maybe more than 1,
or infinitely many)
(b) Suppose A;x = bị is an overdetermined linear system and Aży = b2 is an underdetermined linear
|A)x = bị ,
| A2y = b2
system. How many solutions do you expect the combined system
to have? (Note that
the systems of variables x and y are unrelated.)
(c) Suppose A;x = bị and A,x = b, are linear systems of n equations in the same collection of n
variables r1,..., Tn. How many solutions do you expect the combined system
[A]
|A2]
to
b2
have?
(d) Suppose A¡x = bị and A2y = b2 are underdetermined linear systems where bị and bą are m-vectors
for a common m. How many solutions do you expect the combined system [A A]
= bị + b2
to have?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b025f1d-c8f1-4721-94b0-e23bf12ccb3b%2F236d975c-a3ff-47fb-885b-6db6bb9dc711%2Fq1md80b.png&w=3840&q=75)
Transcribed Image Text:Exercise 21.12. This exercise involves the overdetermined and underdetermined ideas when we combine
linear systems. In each case below, if you don't think there is a reasonable guess for the behavior of the
combined linear system (e.g. if it depends on the specific sizes of the underlying systems), you should
explain why rather than giving an unreasonable guess. Accompany your answer with some informal
reasoning (please don't write out anything involving long lists of variables or computations with them).
(a) Suppose A,x = bị and A2y = b, are overdetermined linear systems. How many solutions do you
A¡x = bị
A2y = b2
%3D
expect the combined system
to have? (e.g. 0, 1, finitely many but maybe more than 1,
or infinitely many)
(b) Suppose A;x = bị is an overdetermined linear system and Aży = b2 is an underdetermined linear
|A)x = bị ,
| A2y = b2
system. How many solutions do you expect the combined system
to have? (Note that
the systems of variables x and y are unrelated.)
(c) Suppose A;x = bị and A,x = b, are linear systems of n equations in the same collection of n
variables r1,..., Tn. How many solutions do you expect the combined system
[A]
|A2]
to
b2
have?
(d) Suppose A¡x = bị and A2y = b2 are underdetermined linear systems where bị and bą are m-vectors
for a common m. How many solutions do you expect the combined system [A A]
= bị + b2
to have?
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