Question

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.

Answer 1

Given $V$ is a vector space over the field $K$ .

$u_1,u_2,u_3,u_4$ are linearly independent$(L.I)$ vectors in $V$ .

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

$$a(u_1+u_2)+b(u_3-u_4)+c(u_4+u_1)=0\\ \implies (a+c)u_1+au_2+bu_3+(c-b)u_4=0\\ \implies (a+c)=a=b=(c-b)=0\hspace{1cm}(\because u_1,u_2,u_3,u_4 \,L.I)\\ \implies a=b=c=0$$

Thus,$u_1+u_2,u_3-u_4,u_4+u_1$ are $L.I$ .