Negation in vector space

Dependencies:

1. Vector Space
2. Zeros in vector space

Let $V$ be a vector space over $F$. Let $a\in F$ and $\mathbf{v}\in V$.

1. $-v=(-1)v$.
2. $-(av)=(-a)v=a(-v)$.

Proof

$$0=0v=(1+(-1))v=v+(-1)v\Rightarrow (-1)v=-v$$ $$0=0v=(a+(-a))v=av+(-a)v\Rightarrow (-a)v=-(av)$$ $$0=a0=a(v+(-v))=av+a(-v)\Rightarrow a(-v)=-(av)$$

Dependency for:

1. Condition for being a subspace

Info:

• Depth: 5
• Number of transitive dependencies: 5

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space
5. Zeros in vector space