About definition in Linear Algebra concerning empty subset of vector space

Question

I'm reading Friedberg's linear algebra:

(1) From definition of both "linear combination" and "span": The subset of concern should be nonempty.

(2) From definition of both "linearly dependent" and "basis": The subset of concern is ... just a subset.

But why (1) put additional restriction on the subset concerned?or, maybe, why (2) didn't?

I know that since the basis of $\{0\}$ is $\{\}$, so (2) have to include all subset including the empty one.

And since that the definition of basis use the word "linearly independent", so the definition of "linearly dependent" also include all subset including the empty one.

And, to the definition of "linearly dependent": "A subset S of a vector space V is called linearly dependent if there exist a finite number of distinct vectors ..."

So this means that $0$ is not a finite number?(But I think it's finite) since $\{\}$ has no elements and it's not linearly dependent?

Answer 1

A subset $S$ of a vector space $V$ over a field $\mathbb{F}$ is called linearly dependent if there are vectors $v_1,\dots,v_n$ and scalars $a_1,\dots,a_n$ such that $$ \sum_{k=1}^n a_k v_k = 0. $$ As such, all linearly dependent sets must be nonempty.

On the other hand, if $v$ is any nonzero vector, and $a$ is any nonzero scalar, then $av$ is nonzero. Consequently, the only possible basis for $\{ 0\}$ is itself — but this set is clearly linearly dependent since $1\cdot 0 = 0$. If every vector space is to have a basis, we are left with no other choice but to define $\text{span}(\emptyset) =\{ 0 \}$ (note that the empty-set is vacuously linearly independent).

Answer 2

Regarding whether the span of the empty set "should" be defined to be $\{0\}$: There are at least two ways to look at this.

(i) Rephrase the definition to say the span of $S$ is the smallest subspace containing $S$. That's exactly the same as "the set of all linear combinations of elements of $S$" if $S$ is nonempty, but it applies equally well to the empty set.

(ii) Regard the sum of no vectors at all to be the zero vector. After all, what else should it be? Or consider the natural way to define the sum of a finite set:

def sum(S):
  s=0
  for v in S:
    s=s+v
  return s

Answer 3

For a given non-emtpy subset of a vector space $\mathsf{V}$, the $\mathbf{0_\mathsf{V}}$ is always achievable by just making all the scalars $0$. But if we apply the empty set to the definition of "linear combination", the result is that $\mathbf{0_\mathsf{V}}$ is not a linear combination of vectors in $\{\}$, since we don't have any vector to do it. This new empty-set-included definition will make the empty set very special, which is not what we want.

On the other side, there is no problem about $\{\}$ in the definition of linearly independent, so we don't have to specifically exclude the empty set.

After all some part of math be like that is because people want it to be like that.