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A vector space over a field $F$ is a non-empty set $V$ with a binary operation $\oplus ︎$ on $V$ and a binary function $\otimes :F\times V\to V$ such that the following hold for any $a,b\in F$ and $u,v\in V$

• $\oplus ︎$ is associative and commutative
• There is a identity element $0_V$ with respect to $\oplus ︎$
• $1_F$ is an identity element for $\otimes$
• $\oplus ︎$ has inverses for $0_V$
• $\otimes$ distributes into $\oplus ︎$
• $\left(a{+}_{F}b\right)\otimes v=(a\otimes v)\oplus ︎(b\otimes v)$
• $a\otimes (b\otimes v)=\left(a{\cdot }_{F}b\right)\otimes v$

A subset $W$ of a vector space $V$ over a field $F$ is called a subspace of $V$ if $W$ itself is a vector space over $F$ with the same operations of vector addition and scalar multiplication as $V$. Specifically, $W$ is a subspace if:

1. $0_V\in W.$
2. $\forall u,v\in W,u+v\in W$
3. $\forall u\in W,c\in F,c\cdot u\in W$