Prove-the-theorem-A-non-empty-subset-W-of-a-vector-space-V-F-is-the-subset-of-V-if-and-only-if-W-1-W-2-W-F-and-W-1-W-2-W-

Tinku Tara August 5, 2024 Algebra 0 Comments

Question Number 210297 by Spillover last updated on 05/Aug/24

Prove the theorem. A non empty subset W of a vector space V(F) is the subset of V if and only if αW_1 +βW_2 ∈W ∀α,β ∈ F and W_1 ,W_2 ∈W

$${Prove}\:{the}\:{theorem}. \\ $$$${A}\:{non}\:{empty}\:{subset}\:{W}\:\:{of}\:{a}\:{vector}\:{space}\:{V}\left({F}\right) \\ $$$${is}\:{the}\:{subset}\:{of}\:{V}\:\:{if}\:\:{and}\:{only}\:{if} \\ $$$$\alpha{W}_{\mathrm{1}} +\beta{W}_{\mathrm{2}} \:\in{W}\:\:\forall\alpha,\beta\:\in\:{F}\:\:{and}\:{W}_{\mathrm{1}} ,{W}_{\mathrm{2}} \:\in{W} \\ $$

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Prove-the-theorem-A-non-empty-subset-W-of-a-vector-space-V-F-is-the-subset-of-V-if-and-only-if-W-1-W-2-W-F-and-W-1-W-2-W-

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