In a vector space, the most basic relationship between linearly independent sets and spanning sets is that of the Lemma. Under this definition, the list I gave would not be "integrally linearly independent" because we can get $0$ as $3(2) -2(3) = 0$.) Whereas "spanning set" is extrinsic: whether a set of vectors spans depends on which vector space you are working on. Note that "linearly independent" is intrinsic: it depends on the vectors (and the vector space operations), and only on them. Intuitively, the list is "sufficient" to get you all vectors in $V$ (via linear combinations). One way to think about a spanning set for a vector space is:Ī list of vectors in $V$ is a spanning set if every vector of $V$ is in the span of the list. Intuitively, the list doesn't have any (linear redundancies).Īnother, more intrinsic way of thinking about linearly independent lists is:Ī list of vectors is linearly independent if and only if no vector in the list is a linear combination of the other vectors in the list.
Intuitively, the list is minimal for its span: remove any vector, you get a strictly smaller span. One way to think about a linearly independent list of vectors is:Ī list of vectors is linearly independent if and only if removing any vector from the list will result in a list whose span is strictly smaller than that of the original list.
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