Zero Vector Space Product iff Factor is Zero/Proof 2
Jump to navigation
Jump to search
Theorem
Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms.
Let $\mathbf v \in \mathbf V, \lambda \in F$.
Then:
- $\lambda \circ \mathbf v = \bszero \iff \paren {\lambda = 0_F \lor x = \bszero}$
Proof
The sufficient condition is proved in Vector Scaled by Zero is Zero Vector, and in Zero Vector Scaled is Zero Vector.
The necessary condition is proved in Vector Product is Zero only if Factor is Zero.
$\blacksquare$