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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 = b2 are overdetermined linear systems. How many solutions do you | A;x = bị | A2y = b2 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 = b, is an underdetermined linear A1x = bị system. How many solutions do you expect the combined system to have? (Note that | A2y = b2 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 ¤1,..., In. How many solutions do you expect the combined system |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 [A1 Az]C = b1 +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 = b2 are overdetermined linear systems. How many solutions do you | A;x = bị | A2y = b2 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 = b, is an underdetermined linear A1x = bị system. How many solutions do you expect the combined system to have? (Note that | A2y = b2 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 ¤1,..., In. How many solutions do you expect the combined system |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 [A1 Az]C = b1 +b2 to have?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.5: Iterative Methods For Solving Linear Systems
Problem 23EQ
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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 = b2 are overdetermined linear systems. How many solutions do you | A;x = bị | A2y = b2 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 = b, is an underdetermined linear A1x = bị system. How many solutions do you expect the combined system to have? (Note that | A2y = b2 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 ¤1,..., In. How many solutions do you expect the combined system |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 [A1 Az]C = b1 +b2 to have?
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 = b2 are overdetermined linear systems. How many solutions do you | A;x = bị | A2y = b2 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 = b, is an underdetermined linear A1x = bị system. How many solutions do you expect the combined system to have? (Note that | A2y = b2 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 ¤1,..., In. How many solutions do you expect the combined system |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 [A1 Az]C = b1 +b2 to have?
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