subset $U$ of $V$ is subspace of $V$ if $U$ is also a [[vector space]] **conditions:** only need to test for 1. additive identity $0 \in U$ 2. closed under addition 3. and scalar multiplication Matching intuition: If $V$ is [[finite dimensional|fin dim]] then $\dim U \le \dim V$. usually these are just [[linear span]]s of lists of vectors; [[eigenspace]]s form an important class ![[sum of linear subspaces]] eg set of all [[differentiable]] functions $\mathbb R \to \mathbb{R}$ is a subspace of all functions from $\mathbb{R} \to \mathbb{R}$ # sources [[2015AxlerLinearAlgebraDone|Axler]] 1.32 1.34 1.35 1.36 1.39 2.{26,38}