A subspace is just the span of some vectors that is every point you would reach if you were to take all possible linear combinations.

Take \(v_1, v_2, \dots, v_k \in \mathbb{R}^n\). The subspace they generate is
$$
\text{span}{v_1, v_2, \dots, v_k} = { a_1 v_1 + a_2 v_2 + \cdots + a_k v_k : a_i \in \mathbb{R} }.
$$
That set is guaranteed to:
contain the origin (\(0 = 0 \cdot v_1 + \cdots + 0 \cdot v_k\)),
be closed under addition,
* be closed under scalar multiplication.
In \(\mathbb{R}^3\), this could be a line through the origin, a plane through the origin, or the whole space itself.