Let V C R" be a M-element subset of a vector space R" such that span V = R", and let c :V → R>o be a function. For W C V, we define the cost of W, denoted c(W), to be CW) Σε(ν). vƐW For a basis B* CV of R", we call B* optimal if c(B*) < c(B) for all other bases B of R" with B C V. (Note that we're restricting the bases we're considering to ones that are subsets of V, otherwise the cost would not be defined.) (a) Assume that BC V is a non-optimal basis. Show that there exists a v E B and u e V \ B such that B' = (B \ {v}) U {u} is a basis with c(B') < c(B).

Let V C R" be a M-element subset of a vector space R" such that span V = R", and let c :V → R>o be a function. For W C V, we define the cost of W, denoted c(W), to be CW) Σε(ν). vƐW For a basis B* CV of R", we call B* optimal if c(B*) < c(B) for all other bases B of R" with B C V. (Note that we're restricting the bases we're considering to ones that are subsets of V, otherwise the cost would not be defined.) (a) Assume that BC V is a non-optimal basis. Show that there exists a v E B and u e V \ B such that B' = (B \ {v}) U {u} is a basis with c(B') < c(B).

Name: Show the details please. Let V C R" be a M-element subset of a vector
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Description: Show the details please. Let V C R" be a M-element subset of a vector space R" such that span V = R", and letc:V → R>o be a function. For W CV, we define the cost of W, denoted c(W), to bec(W) = c(v).vɛWFor a basis B*C V of R", we call B* optimal if

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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