# Concept derivation guides

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Concept guide



## [Linear combinations and span: deciding membership using finite coefficient lists as witnesses](/en/derivations/derivation-mko-linear-combination-span-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-linear-combination-span)

(1) In ℝ², with v₁=(1,2) and v₂=(3,−1), decide whether b=(7,0) lies in span{v₁,v₂}: first prove that the span of two vectors is exactly {c₁v₁+c₂v₂}, then solve for the coefficients, prove they are unique, and rebuild b from the witness. (2) In ℝ³, with u₁=(1,0,1) and u₂=(0,1,1), prove that t=(2,3,4) is not in span{u₁,u₂} while t'=(2,3,5) is: having three coordinates does not decide membership.

Linear combinations and span

derivation-mko-linear-combination-span-01





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