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Question Number 88552 by M±th+et£s last updated on 11/Apr/20

let W_1 ,W_2 ,....,W_n  be subspaces of a vector  space V over a field (F,+,.)  prove that:  (1) W_1 ∩W_2 ∩....∩W_n  a subspace  of the vector space V  over (F,+,.).  (2)W_1 +W_2 +....+W_n  is subspace of the  vector space V over (F,+,.)

$${let}\:{W}_{\mathrm{1}} ,{W}_{\mathrm{2}} ,....,{W}_{{n}} \:{be}\:{subspaces}\:{of}\:{a}\:{vector} \\ $$$${space}\:{V}\:{over}\:{a}\:{field}\:\left({F},+,.\right) \\ $$$${prove}\:{that}: \\ $$$$\left(\mathrm{1}\right)\:{W}_{\mathrm{1}} \cap{W}_{\mathrm{2}} \cap....\cap{W}_{{n}} \:{a}\:{subspace} \\ $$$${of}\:{the}\:{vector}\:{space}\:{V}\:\:{over}\:\left({F},+,.\right). \\ $$$$\left(\mathrm{2}\right){W}_{\mathrm{1}} +{W}_{\mathrm{2}} +....+{W}_{{n}} \:{is}\:{subspace}\:{of}\:{the} \\ $$$${vector}\:{space}\:{V}\:{over}\:\left({F},+,.\right) \\ $$

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