A) W is a sub space of V
given that V =R^4 and W= [(x1, x2, x3, 0)|x1 ,x2, x3 E..R]
W is a subspace of V because of the following.
let (x1, x2, x3, 0), (y1, y2, y3, 0) ∈ W and a, b ∈ V
now
a(x1, x2, x3, 0)+b (y1, y2, y3, 0)= (a x1, a x2, a x3, 0)+(b y1, b y2,b y3, 0)
= (a x1+ b y1 , a x2+ b y2 , a x3+ b y3 ,0+0)=
(a x1+ b y1 , a x2+ b y2 , a x3+ b y3 ,0+0)
LHS =RHS
∈W
since (a x1+ b y1 , a x2+ b y2 , a x3+ b y3 ) are all elements of R
W is closed under vector addition and scalar multiplication.
Hence W is a sub space of V.
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