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jwqwerty

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1. Homework Statement

If set A={u,v,w} ⊂ R^n is linearly independent, is B={u-v, u+w, v+w}⊂ R^n linearly independent?

2. Homework Equations

3. The Attempt at a Solution

Since A is linearly independent, there exist no all non-zero scalars a1, a2, a3 such that a1*u+a2*v+a3*w=0. Or I can say that since A is linearly independent, a1=a2=a3=0. Then to determine whether B is linearly indepedent, I think I need to determine whether b1=b2=b3=0 for b1*(u-v)+b2*(u+w)+b3*(v+w)=0. But if I do so, I find no connection between a1, a2, a3 and b1, b2 ,b3. How can I complete this proof?

If set A={u,v,w} ⊂ R^n is linearly independent, is B={u-v, u+w, v+w}⊂ R^n linearly independent?

2. Homework Equations

3. The Attempt at a Solution

Since A is linearly independent, there exist no all non-zero scalars a1, a2, a3 such that a1*u+a2*v+a3*w=0. Or I can say that since A is linearly independent, a1=a2=a3=0. Then to determine whether B is linearly indepedent, I think I need to determine whether b1=b2=b3=0 for b1*(u-v)+b2*(u+w)+b3*(v+w)=0. But if I do so, I find no connection between a1, a2, a3 and b1, b2 ,b3. How can I complete this proof?

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