> [!summary] A subspace is a subset of a vector space. > **Subspace Test:** $\vec{0}\in W$ $\vec{v}+\vec{u} \in W$ $c\vec{u} \in W$ >[!info]+ Read Time **⏱ 2 mins** # Definition A [[Subsets|subset]] ($W$) of a [[Vector Spaces|vector space]] ($V$) is called a subspace of $V$, if $W$ is a vector space. Meaning that $W$ satisfied [[Vector Addition & Scalar Multiplication|scalar multiplication and vector addition]]. If it does satisfy these conditions, then $W$ is **closed under [[Vector Addition & Scalar Multiplication|vector addition and scalar multiplication]].** > [!note]+ Subspace Diagram ![[sp_1.png|300]] Visual example of a subspace of a vector space ## Subspace Test > [!warning] Assumptions To prove $W$ is a subspace of $V$, it must satisfy the 3 tests. Before showing the tests, assume the following: > - The [[Vector Spaces|vector space]] $V$ is over a [[Field|field]] $F$ > - $\vec{u},\vec{v}$ are vectors in $W$ ($\vec{u},\vec{v}\in W$) > - $c$ a [[Field|field of scalars]] ($c\in F$) After assuming those assumptions, then $W$ is a subspace of $V$ only if all 3 of the following properties are satisfied: 1. There exists a [[Zero Vector|zero vector]] in $W$ so $\vec{0}\in W$ 2. Closed under addition so $\vec{v}+\vec{u} \in W$ 3. Closed under scalar multiplication, so $c\vec{u} \in W$ > [!info]- Where are the properties derived from? Technically, from [[Vector Spaces|vector spaces]] there are 10 properties/axioms when it's assumed vector addition and scalar multiplication are valid. > But the 3 properties for subspaces guarantee that all 10 axioms are followed. # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/n3GRsHYcchQ?si=KdSyZRAStvarayg4" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>