Linear independance and span (Definition)

[sid9221]

Would I be correct in saying that:

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

With S being a subset of V.

 

[Number Nine]

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, ... s_n are linearly independent, then the equality a₁s₁ + a₂s₂ + ... + a_ns_n = 0 has only the trivial solution a₁ = a₂ = ... = a_n = 0 (where a₁, a₂,... a_n 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).

 

[Fredrik]

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").