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).
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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Transcribed Image Text: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 CV, we define the cost of W, denoted c(W), to be
c(W) = c(v).
vɛW
For a basis B*C V 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).
(b) Show that the following algorithm outputs an optimum basis.
Set B := Ø.
Sort the vectors of V so that c(v1) < c(v2) < · · < c(Um).
For i = 1 to m do:
If BU {v;} is linearly independent, then update B :=
BU{v;}.
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