#math/linear-algebra Let $V$ be a vector space. A subset $U$ of $V$ is called a `subspace`, or `linear subspace`, of $V$ if $U$ is also a [[1B1 - Vector Space|vector space]] (using the same addition and scalar multiplication as on $V$). Formally, $U$ is a subspace of $V$ if and only if the following hold true: - Has additive identity: $0 \in{} U$ - Is closed under addition: $u, w \in{} U \implies u + w \in{} U$ - Is closed under scalar multiplication: $a \in{} F \text{ and }u \in{} U \implies au \in{} U$ All other vector-space axioms are inherited from $V$. For example, by closure under scalar multiplication, additive inverses follow because $-u=(-1)u\in U$. The remaining naturally follow.