Statement:-
The linear
sum of two subspaces of a vector space is also a subspace of same vector space.
Proof:-
Let W1
and W2 be two subspaces of a vector space V.
Since, W1
and W2 are non-empty set.
∴
W1+W2≠ϕ
Let α, β∊W1+W2 and a, b∊F then
α=α1+α2
for some α1∊W1 and α2∊W2
and β=β1+β2 for some β1∊W1 and β2∊W2
Now, aα+bβ= a(α1+α2)+b(β1+β2)
=aα1+aα2+bβ1+bβ2
=(aα1+aα2)+(aα2+bβ1)
Since W1 is a subspace.
Hence a,b∊F,
α1,β1∊W1 ⇨ aα1+bβ1∊W1
Also W2 being a subspace.
a, b∊F and α2,β2∊W2 ⇨
aα2+bβ2∊W2
Thus aα+bβ=(aα1+bβ1)+(aα2+bβ2)∊W1+W2
Since a,bєF and α,βєW1+W2 ⇨
aα+bβєW1+W2
∴ W1+W2 is a
subspace of V.
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