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
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.
0 comments:
Post a Comment