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