Subspace

Definition

A subspace is a Subset $W$ of a Vector Space $V$ that is itself a vector space under the same operations of addition and scalar multiplication defined on $V$.

Core Idea

Not every Subset of a space is a subspace. A subset becomes a subspace only if it satisfies the vector-space axioms. (In practice this requirement collapses to three key conditions listed below)

Subspace Test

A subset $V\subseteq V$ is a subspace iff:

1. Zero Vector is in $W$
2. Closed under Addition
3. Closed under Scalar Multiplication

If these 3 hold, the rest of the Vector Space axioms come “for free” because they are inherited from $V$.

Subspaces of ${\mathbb{R}}^n$

Below are common examples of subspaces of ${\mathbb{R}}^n$

• The Zero Vector: $\{0\}$
• Lines through the origin in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$
• Planes through the origin in ${\mathbb{R}}^3$
• The Vector Span of any collection of vectors.

Key Point: In ${\mathbb{R}}^n$ only structures that pass through the origin can be subspaces.