Let p,q∈R and (a1,a2,a3),(b1,b2,b3)∈W.
Consider
p(a1,a2,a3)+q(b1,b2,b3)=(pa1+qb1,pa2+qb2,pa3+qb3) Now
pa2+qb2+pa3+qb3=p(a2+a3)+q(b2+b3)=0 So p(a1,a2,a3)+q(b1,b2,b3)∈W, and hence W is a subspace of R3.
Next, let W1={(x2,x2,0)} and W2={(x2,2x2,0)}.
Suppose
s∈W∩W1 Then every element of s is of the form (x1,x2,x3) where x3=0, x2=−x3=0 and x1=x2=0. Hence s={(0,0,0)}.
Also, if (a,b,c)∈R3 then it can be written as
(a,b,c)=(a−b−c,−c,c)+(b+c,b+c,0) So R3=W⨁W1.
Similarly, suppose t∈W∩W2. Then every element of t is of the form (y1,y2,y3) where y3=0, y2=−y3=0 and y1=y2=0.
Also, if (a,b,c)∈R3 then it can be written as
(a,b,c)=(22a−b−c,−c,c)+(2b+c,b+c,0) So R3=W⨁W2.
Clearly, W1=W2.
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