(a) Consider a system Ax = b. If the system is consistent, which fundamentalsubspace of A contains b? Why?(b) If the system Ax = b is inconsistent, then b must have a component in whichfundamental subspace of A? Why?(c) The previous two parts imply that the vector b may be decomposed as b =Va + Vb, where va is the component of b in the fundamental space from part(a), and vb is the component of b in the fundamental space from part (b). Whatis the relationship between va and vh? Why?(d) Our goal is to find a vector Âå such that Ax minimizes the distance from b. Howdoes Ax relate to the vectors va and v, above?This is thequestion

Let the given system be Ax=b. Assume that the system has no solution. Then we have to find the…

Answered: d) Our goal is to find a vector ÂY such…

d) Our goal is to find a vector ÂY such that Ax minimizes the distance from b. How does Aâx relate to the vectors va and vþ above?

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...

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Concept explainers

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.

Question

(a) Consider a system Ax = b. If the system is consistent, which fundamental
subspace of A contains b? Why?
(b) If the system Ax = b is inconsistent, then b must have a component in which
fundamental subspace of A? Why?
(c) The previous two parts imply that the vector b may be decomposed as b =
Va + Vb, where va is the component of b in the fundamental space from part
(a), and vb is the component of b in the fundamental space from part (b). What
is the relationship between va and vh? Why?
(d) Our goal is to find a vector Âå such that Ax minimizes the distance from b. How
does Ax relate to the vectors va and v, above?
This is the
question

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