# Concept derivation guides

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Concept guide



## [Linear independence and bases: proving an ordered basis of ℝ² by exact elimination, with two counterexamples that separate the two obligations](/en/derivations/derivation-mko-linear-independence-basis-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-linear-independence-basis)

In ℝ² let v_1=(1,2), v_2=(3,5), v_3=(1,1), w=(4,7). (1) Prove by exact elimination that (v_1,v_2) is linearly independent; (2) find the coefficients for an arbitrary (p,q), proving span(v_1,v_2)=ℝ², so that (v_1,v_2) is an ordered basis; (3) find the unique coordinates of w; (4) prove that (v_1) is independent but does not span ℝ², while (v_1,v_2,v_3) spans ℝ² but is not independent, and w has more than one representation in it.

Linear independence and bases

derivation-mko-linear-independence-basis-01





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