Subspaces

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.