Span of the trivial vector space I was wondering is the span of the trivial vector space the {0} vector in $\Bbb R^2$ for example just 1 vector $(0,0)$ or can it be any vectors in $\Bbb R^2$ since we can pick $t \in\Bbb R$ to be zero so in this case there infinite spanning vectors ?
 A: The span of a set of vectors $K$ in a vector space $V$ is the collection of linear combinations of them, which means finite sums $a_1v_1+...+a_rv_r$ where the $a_k$ are any scalars and each $v_k \in K.$ For your question $K$ is the set $\{0\}$ having only one vector in it, so the "sums" just mentioned have only one term $a_1v_1$ where $v_1=0,$ the zero vector of $V.$ One of the definitions of scalar multiplication of a scalar by a vector says that if the vector is $v=0$ then $a_1v=0,$ i.e. again the zero vector. So in this case there are no other vectors in the span than the zero vector. (You can't "reach" other vectors in $V$ by multiplying the zero vector by a scalar, for the reason just mentioned.)
A: I think it just 1 vector since The 0 vector has no basis so it can't be expressed as linear combination of independent vectors, so by that fact the vectors that can represent it is only itself.
A: One way of describing the span of a given set is the intersection of all subspaces of a given vector space containing said set; because all subspaces spaces contain the empty set, we need to take the intersection of all subspaces.  Every subspace contains zero (this is an axiom), so their intersection is Span$(\emptyset)=0$.
EDIT:  Replace "empty set" with "zero vector" and "$\emptyset$" with "$0$", to be consistent with what was asked, otherwise the ideas don't change.
