{v1} linearly independent if and only if
the span {v1} is a linex1v1=0 has only the trivial solution x1=0 and v1 is not the hero vector
A set of vectors in a vector space, V, is said to be linearly independent if none of the vectors in the set can be expressed as a linear combination of the others. That is, if we have vectors v1, v2, …, vn, then they are linearly independent if the only solution to the equation:
c1v1 + c2v2 + … + cnvn = 0
is the trivial solution where c1, c2, …, cn are all equal to zero.
The statement vectors v1, v2, …, vn are linearly independent if and only if is incomplete. To complete it, we need to know what property or condition we are comparing linearly independence to. For example, we could say vectors v1, v2, …, vn are linearly independent if and only if they do not span the vector space V or vectors v1, v2, …, vn are linearly independent if and only if they have maximal cardinality.
Without specifying what we are comparing linear independence to, we cannot provide a complete answer to the question.
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