1. The problem statement, all variables and given/known data(adsbygoogle = window.adsbygoogle || []).push({});

Suppose that {v1, v2, v3} is a linearly independent subset of R^M. Show that the set {v1, v1 + v2, v1 + v2 + v3} is also linearly independent.

3. The attempt at a solution

So I know that {v1, v2, v3} is contained in R^M. And that since the set is linearly independent, the only solution to the vector equation a1v1 + a2v2 + a3v3 = 0 is a1 = 0, a2 = 0, and a3 = 0.

It is true to say that the vector {v1, v1 + v2, v1 + v2 + v3} is linearly independent because for the first vector to be linearly independent, the dimension (M) would have to be greater than the number of vectors (which in this case is three). We know from a theorem that if p (the number of vectors) is greater than m (the dimensional space) then the set is linearly dependent. In this case, for the second set to be linearly independent, M would have to be equal to or less then the number of vectors. However, since the number of vectors is still the same in both the first and second set, the set is linearly dependent...

Correct?

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# Homework Help: Linear dependence of vectors

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