Every Subset of a Linearly Independent Set is also Linearly Independent Proof

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A proof that every subset of a linearly independent set is also linearly independent.
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  2. Linear Independence
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Автор

Can you make a proof on any set of vectors containing a linearly dependent subset is linearly dependent.

septembabes
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is the converse also true? If a set S has a linearly independent subset, then S itself is also linearly independent?

tabmax
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Hi, Does this proof work? :

I started by assuming that S' is not linearly independent. Then ∃x∈V :
x = k1v1 + k2v2 + k3v3 + ... + knvn
where v1, v2, .., vn ∈ S'

=> k1v1 + k2v2 + k3v3 + ... + knvn + (- 1)x = 0
But x ∈ S. So this means we have gotten the zero vector as a linear combination of the other vectors in S with not all the scalars being zero, which makes S dependent, which is a contradiction. Hence our assumption is wrong and S' is linearly Independent.

legendarynoob