**Definition:** A subset \(W\) of a vector space \(V\) is a *subspace* if \(W\) is itself a vector space under the same operations of addition and scalar multiplication defined on \(V\).

**Subspace test:** A nonempty subset \(W \subseteq V\) is a subspace if and only if:
1. **Closed under addition:** If \(\mathbf{u}, \mathbf{v} \in W\), then \(\mathbf{u} + \mathbf{v} \in W\)
2. **Closed under scalar multiplication:** If \(\mathbf{v} \in W\) and \(c\) is a scalar, then \(c\mathbf{v} \in W\)

(These two conditions guarantee \(\mathbf{0} \in W\) and that additive inverses exist.)

**Examples:**
- \(\{\mathbf{0}\}\) is a subspace of every vector space (the *trivial subspace*)
- Every vector space \(V\) is a subspace of itself
- In \(\mathbb{R}^3\), any line or plane through the origin is a subspace
- The set \(\{(x,y,0) : x,y \in \mathbb{R}\}\) is a subspace of \(\mathbb{R}^3\)

**Non-example:** The set \(\{(x,y,1) : x,y \in \mathbb{R}\}\) is NOT a subspace of \(\mathbb{R}^3\) because it does not contain the zero vector.