Given V is a vector space over the field K .
u1,u2,u3,u4 are linearly independent(L.I) vectors in V .
Now, let for all a,b,c∈K , consider the linear combination
a(u1+u2)+b(u3−u4)+c(u4+u1)=0⟹(a+c)u1+au2+bu3+(c−b)u4=0⟹(a+c)=a=b=(c−b)=0(∵u1,u2,u3,u4L.I)⟹a=b=c=0 Thus,u1+u2,u3−u4,u4+u1 are L.I .
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