Why does the definition of a linear subspace asks for non-emptiness?

Question

If a subset $W$ of a vector space $V$ is closed under addition and scalar multiplication, then

$$ 0 \cdot v = 0 \in W $$

So we have non-emptiness. But all books I have about linear algebra ask for non-emptiness nonetheless. Why?

Answer 1

We need to have the $0$ vector in any subspace so it can not be empty.

Answer 2

The condition merely states that the zero vector (however it's defined by vector space $V$) is also in subspace $W$. At minimum, any subspace must have at least one element, namely the zero vector. This fulfills both the non-emptiness criteria and the equation you cited above.

Answer 3

If we have a vector space $S $, then it is non-empty by definition, so some $u\in S $ and hence by the scalar closure, we also have $0u={\bf 0}$. This means non-emptiness and closure under scalar multiplication imply that zero vector must be in the space.

On the other hand, closure under addition and scalar multiplication defined/satisfied on an empty set would give nothing useful.

You seem to confuse non-emptiness of a set with the condition that zero be an element of that set. In general a set can be non-empty and not contain zero. For example $S=\{1,5\}$ is non-empty (of course it is not a vector space either).

Your books require non-emptiness of the set, not the space. So note the difference between the words ${\bf set} $ and ${\bf space}$.