Basis of $\{0\}$ set I am solving Linear Algebra and having a trivial doubt .
Is W ={∅} i.e. an empty set a basis of ={0} ?
I have read some solutions regarding the above and they imply that since W contains no vector , W  by definition is linearly independent .
I am not sure how W spans V. Can anyone explain this to me ?
"Every vector space has a basis."

Is the above statement true ?
 A: If $V=\{0\}$ - a trivial vector space, then $W=\emptyset$ is its basis. (Note: not $W=\{\emptyset\}$ but $W=\emptyset=\{\}$.)
This empty set spans $V$ because any sum of no addends is taken, by convention, to be zero (in this case a zero vector).
This also matches the alternative definition of a span: "the smallest subspace containing all the vectors from the set". In this case, any subspace (including $\{0\}$) vacuously contains all the vectors from the empty set.
A: The standard convention for any binary operation with a neutral element is that the "empty operation" gives the neutral element. Some specific examples:

*

*The empty sum is $0$: $\sum_{x\in\emptyset}x=0$.

*The empty product is $1$: $\prod_{x\in\emptyset}x=1$.

*The empty union is the empty set: $\bigcup_{A\in\emptyset}A=\emptyset$.

Important for your question is the first one in the list: A linear combination of vectors in the empty set is the zero vector:
$$\sum_{v\in W}a_vv=\sum_{v\in\emptyset}a_v v=0.$$
And thus $W=\emptyset$ (not $\{\emptyset\}$!) spans the trivial vector space. It is also linearly independent, since there is no non-trivial linear combination which results in the zero vector.
