MATH 307 Lecture 17: Lecture17 Feb 12, 2020 11_59_09 AM
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0 x of solutions sothereis ito s 1 vc i. Vi vz vz arelinearly dependent why rc i c iag. So whenken reduce vc 81 1tokc 8 andcheck it has afreevariable column. If ithas then vi uk are linearlydependent or pivots c k. Def spanwi uk is a subspace formedby all possiblelinear combinations ofvectors. Span114 a14 la er e423 fatg t a1 i a cite. Rs 114 a a ger span fig 1. Def a collection vi vk is a basisforsubspace 5 if 1spanvi vn s z vi vkarelinearlyindepend. Basisfor423 f f thisis a basisfora subspace 5 11 is not abasisfor1123since itdoesnotspan423. Fact vi vk is abasisfor5 then any vector ues can bewritten as a linearcombination of vi vk uniquely proofidea suppose not then thereis avectorves tclerk d vit. Vicnt contradictsthat ivi vk are linearlyindependent property2ofbasis tdkuk some g t dj.