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?
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.