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Exercise 18

Avg difficulty 3.0

Where ⟨β⃗1,…,β⃗n⟩\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle is a basis, show that in this equation

c1β⃗1+⋯+ckβ⃗k=ck+1β⃗k+1+⋯+cnβ⃗nc_{1}\vec{\beta}_{1}+\dots+c_{k}\vec{\beta}_{k}=c_{k+1}\vec{\beta}_{k+1}+\dots+c_{n}\vec{\beta}_{n}

each of the cic_{i}’s is zero. Generalize.

Hefferon, Linear Algebra, Vector Spaces, Basis, Exercise 18

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.