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Linear independance and span (Definition)

  1. May 19, 2012 #1
    Would I be correct in saying that:

    If Span(S)≠0 then S is linearly independant.
    If Span(S)=0 then S is linearly dependant.

    With S being a subset of V.
     
  2. jcsd
  3. May 19, 2012 #2
    No. Span describes the set of all vectors in V that are linear combinations of vectors in S, it is entirely separate from linear independence. Linear independence means (there are various equivalent definitions) the following...

    If the vectors s1, s2, ... sn are linearly independent, then the equality a1s1 + a2s2 + ... + ansn = 0 has only the trivial solution a1 = a2 = ... = an = 0 (where a1, a2,... an are scalars). Equivalently, none of the vectors can be expressed as a linear combination of the others.

    Take, for example, the vectors [1 0 0], [0 1 0], and [0 0 1] in R3. These vectors are linearly independent, and yet they span all of R3 (the term for such a set of vectors is a basis for the space R3).
     
    Last edited: May 19, 2012
  4. May 20, 2012 #3

    Fredrik

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    Suppose e.g. that S is a subset of ##\mathbb R^2## that contains two points on the same line through the origin. For example, ##S=\{(0,1),(0,2)\}##. Then S is linearly dependent, and span S is that line, so span S is neither ∅ nor {0}. (I don't know which of those you meant by "0").
     
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