Suppose that S={v1, v2, v3} is a linearly independent set of vectors in a vector space V. How do you show that T={w1, w2, w3} is also linearly independent , where w1=v1+v2+v3, w2=v2+v3, and w3=v3?

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- March 18th 2009, 06:45 PMantmanlinearly independent
Suppose that S={v1, v2, v3} is a linearly independent set of vectors in a vector space V. How do you show that T={w1, w2, w3} is also linearly independent , where w1=v1+v2+v3, w2=v2+v3, and w3=v3?

- March 18th 2009, 07:17 PMReckoner
Assume the contrary, and suppose that is linearly dependent. Then there must be some not all zero, such that

Since are linearly independent, we have

But this contradicts the requirement that are not all zero. Hence, our initial assumption is false, and is linearly independent.