I am very confused with this proof, if someone could help meout I would really a

Question

I am very confused with this proof, if someone could help meout I would really appreciate it!!
If S= {v1, v2,..., v3} and is a linearly independent set ofvectors in a vector space V, how do I prove that T={w1, w2, w3} isalso linearly independent where w1=v1+v2+v3, w2=v2+v3, andw3=v3? I am very confused with this proof, if someone could help meout I would really appreciate it!!
If S= {v1, v2,..., v3} and is a linearly independent set ofvectors in a vector space V, how do I prove that T={w1, w2, w3} isalso linearly independent where w1=v1+v2+v3, w2=v2+v3, andw3=v3?

Explanation / Answer

If w1 w2 w3 are linearly independent then the only way to have a1*w1 +a2*w2+a3*w3 =0 (**) is if a1=a2=a3=0 but we can rewrite (**) as a1(v1+v2+v3)+a2(v2+v3)+a3(v3)=0 which can be rewritten as a1v1+(a1+a2)v2+(a1+a2+a3)v3=0 but v1,v2, v3 are linearly independent which means that we have tohave a1=0 a1+a2=0 and a1+a2+a3=0 It is pretty easy easy to see from these equations thata1=a2=a3=0 Hence w1,w2,w3 are linearly independent

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