Let V be a
vector space over the field F the linear sum of two subspaces W1 and
W2 of V written as (W1+W2) and is defined as W1+W2={α1+α2:α1єw1,α2єw2} which shows that each element 0f (W1+W2)
is expressible as sum of an element of W1 and an element of W2.
Also,
W1⊆W1+W2 and W2⊆W1+W2
Since,
if α∊W1 then
α=α+0,where α∊W1 and 0∊W2
⇨ α∊W1+W2
∴ α∊W1
⇨
α∊W1+W2
⇨
W1⊆W1+W2
Similarly,
W2⊆W1+W2
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