Question #144981
Show that if u1 , u2, u3, u4 are linearly
independent vectors in a vector space V over
a field K, then u 1+ u2, u3— u4, u4+ u1 are
also linearly independent.
1
Expert's answer
2020-11-19T17:10:51-0500

Given VV is a vector space over the field KK .

u1,u2,u3,u4u_1,u_2,u_3,u_4 are linearly independent(L.I)(L.I) vectors in VV .

Now, let for all a,b,cKa,b,c\in K , consider the linear combination


a(u1+u2)+b(u3u4)+c(u4+u1)=0    (a+c)u1+au2+bu3+(cb)u4=0    (a+c)=a=b=(cb)=0(u1,u2,u3,u4L.I)    a=b=c=0a(u_1+u_2)+b(u_3-u_4)+c(u_4+u_1)=0\\ \implies (a+c)u_1+au_2+bu_3+(c-b)u_4=0\\ \implies (a+c)=a=b=(c-b)=0\hspace{1cm}(\because u_1,u_2,u_3,u_4 \,L.I)\\ \implies a=b=c=0

Thus,u1+u2,u3u4,u4+u1u_1+u_2,u_3-u_4,u_4+u_1 are L.IL.I .


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