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I was asked by someone today to explain the notion of linear independence of a set of vectors and I would just like to check that I explained it correctly.

A set of vectors [itex] S[/itex] is said to be linearly dependent if there exists distinct vectors [itex] \mathbf{v}_{1}, \ldots , \mathbf{v}_{m} [/itex] in [itex] S[/itex] and scalars [itex] c_{1},\ldots c_{m}[/itex], not all of which are zero, such that [tex] c_{1}\mathbf{v}_{1} + \cdots + c_{m}\mathbf{v}_{m} = \sum_{i=1}^{m} c_{i}\mathbf{v}_{i} = \mathbf{0} [/tex]

What this means is that at least one vector in [itex]S[/itex] can be completely specified in terms of the other vectors in the set and hence it is dependent on the particular form of those vectors. However, if the only case for which [itex]\sum_{i=1}^{m} c_{i}\mathbf{v}_{i} = \mathbf{0}[/itex] is the trivial case, in which [itex] c_{i} = 0 \; \forall \; i=1, \ldots , m[/itex], then the set is said to be linearly independent, as none of the vectors contained within it can be specified in terms of the other vectors in [itex]S[/itex].

Is this a valid description of the concept?

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# Linear dependence of vectors

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