Subspace

Definition

Let

• $V$ : Vector space
• $W\subset V$

If $W$ is a vector space under the addition and scalar multiplication defined on $V$

Then $W$ is a subspace of $V$

Theorem

Let

• $V$ : Vector space
• $W\subset V$

Then $W$ is a subspace of $V$

If and only if

• $\mathbf{u},\mathbf{v}\in W⟹\mathbf{u}+\mathbf{v}\in W$
• $k\text{ scalar},\mathbf{u}\in W⟹k\mathbf{u}\in W$