Problem 1 (Changing the generating set of a span) This problem has two parts: (a) Show that if v is a linear combination of v1, ..., Vm and each vi is a linear combination of w1, ..., Wn, then v is a linear combination of W1, ..., wn. (b) Suppose v1, ..., Vm are vectors that span a vector space V. Let w₁, ..., wn Є V be any other elements. Suppose that each v¿ can be written as a linear combination of w₁, ..., Wn. Prove that the vectors w₁, ..., Wn also span V. Challenge: If it so happens that m < n, can you also prove that out of the n vectors w₁,..., Wn, you can pick m of them that will still span V?
Problem 1 (Changing the generating set of a span) This problem has two parts: (a) Show that if v is a linear combination of v1, ..., Vm and each vi is a linear combination of w1, ..., Wn, then v is a linear combination of W1, ..., wn. (b) Suppose v1, ..., Vm are vectors that span a vector space V. Let w₁, ..., wn Є V be any other elements. Suppose that each v¿ can be written as a linear combination of w₁, ..., Wn. Prove that the vectors w₁, ..., Wn also span V. Challenge: If it so happens that m < n, can you also prove that out of the n vectors w₁,..., Wn, you can pick m of them that will still span V?
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:Problem 1 (Changing the generating set of a span) This problem has two parts:
(a) Show that if v is a linear combination of v1, ..., Vm and each vi is a linear combination of w1, ..., Wn, then
v is a linear combination of W1, ..., wn.
(b) Suppose v1, ..., Vm are vectors that span a vector space V. Let w₁, ..., wn Є V be any other elements.
Suppose that each v¿ can be written as a linear combination of w₁, ..., Wn. Prove that the vectors w₁, ..., Wn
also span V.
Challenge: If it so happens that m < n, can you also prove that out of the n vectors w₁,..., Wn, you can
pick m of them that will still
span V?
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