Linear Algebra 6c: Second Definition of Linear Dependence Fixed!

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  2. Linear Algebra (Field Of Study)
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Definition 1 for a linearly dependent set of vectors: A set of vectors is linearly dependent if one of the vectors is in the span of the other vectors.
Specifically if the set of vectors is { v_1, v_2, ..., v_k }, then by relabeling the vectors so that the 'extra' vector appears at the end of the list, we could say v_k ∈ span { v_1, v_2, ... v_k-1 }.

Equivalent definition 2: A set of vectors is linearly dependent if there exists a nontrivial linear combination in the set that equals the zero vector.

The two definitions are clear and easy to apply when the set of vectors contains 2 or more vectors. In the case of a set with a single vector, definition 2 seems easier to use. Thus a set containing a single nonzero vector is linearly independent, and the set { 0 } is dependent.

What about a set with no vectors, i.e. the empty set Ø . Is Ø linearly dependent or independent? I am not sure. And what is the span of the empty set, since there are no vectors to take a linear combinations of?

maxpercer
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Talking about the second definition of linear dependence that additional assumption must be given that there are no zero vectors involved.

sammao
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The first definition could be described as similarity and the second definition could be described as symmetry, right?

judgeomega