Vectors are Left Cancellable
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Theorem
Let $\struct {\mathbf V, +, \circ}$ be a vector space over $\GF$, as defined by the vector space axioms.
Then every $\mathbf v \in \struct {\mathbf V, +}$ is left cancellable:
- $\forall \mathbf a, \mathbf b, \mathbf c \in \mathbf V: \mathbf c + \mathbf a = \mathbf c + \mathbf b \implies \mathbf a = \mathbf b$
Proof
Utilizing the vector space axioms:
\(\ds \mathbf c + \mathbf a\) | \(=\) | \(\ds \mathbf c + \mathbf b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf a + \mathbf c\) | \(=\) | \(\ds \mathbf b + \mathbf c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf a\) | \(=\) | \(\ds \mathbf b\) | Vectors are Right Cancellable |
$\blacksquare$