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Let ℬ = {v1, v2, ..., vn} a basis of a vector space V, then 𝓑 is a maximal set of linear independent vectors of V. Proof. By definition the vectors {v1, v2, ..., vn} are linear independent; we must show that for any other vector v ∈ V, the vectors v1, v2, ..., vn, v are linear dependent. Since ℬ spans V, there exist λ1, ..., λn ∈ 𝕂, such that v = λ1v1 + ... + λnvn thus we have v − λ1v1 − ... − λnvn = O which is a linear dependence relation between the vectors v1, v2, ...,vn (the coefficient of v is 1, which is not null). □
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Sat, 08 Apr 2023 15:45 GMT