Given a vector s (v₁),e m, and let where |J|= n. T uch that LnI= e same subspace

Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.3: Subspaces Of Vector Spaces
Problem 45E: Consider the vector spaces P0,P1,P2,...,Pn where Pk is the set of all polynomials of degree less...
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Given a vector space E, let (u)iel be
any finite linearly independent family in E, where I = m, and let (vj)jes be any finite family
such that every u, is a linear combination of (vj)jes, where |J|= n. Then there exists a set
L and an injection p: L→ J (a relabeling function) such that LnI = 0, |L|=n-m, and
the families (u)ier U (Up())IEL and (vj)jes generate the same subspace of E. In particular,
m ≤n.
Transcribed Image Text:Given a vector space E, let (u)iel be any finite linearly independent family in E, where I = m, and let (vj)jes be any finite family such that every u, is a linear combination of (vj)jes, where |J|= n. Then there exists a set L and an injection p: L→ J (a relabeling function) such that LnI = 0, |L|=n-m, and the families (u)ier U (Up())IEL and (vj)jes generate the same subspace of E. In particular, m ≤n.
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