Vector Augend plus Addend equals Augend implies Addend is Zero

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $V$ be a vector space over a field $F$.

Let $\mathbf a, \mathbf b \in V$.

Let $\mathbf a + \mathbf b = \mathbf a$.


Then:

$\mathbf b = \bszero$

where $\bszero$ is the zero vector of $V$.


Proof

\(\ds \mathbf a + \mathbf b\) \(=\) \(\ds \mathbf a\)
\(\ds \leadsto \ \ \) \(\ds \paren {-\mathbf a} + \paren {\mathbf a + \mathbf b}\) \(=\) \(\ds \paren {-\mathbf a} + \mathbf a\)
\(\ds \leadsto \ \ \) \(\ds \paren {\paren {-\mathbf a} + \mathbf a} + \mathbf b\) \(=\) \(\ds \paren {-\mathbf a} + \mathbf a\) Vector Space Axiom $\text V 2$: Associativity
\(\ds \leadsto \ \ \) \(\ds \bszero + \mathbf b\) \(=\) \(\ds \bszero\) Vector Space Axiom $\text V 4$: Inverses
\(\ds \leadsto \ \ \) \(\ds \mathbf b\) \(=\) \(\ds \bszero\) Vector Space Axiom $\text V 3$: Identity

$\blacksquare$


Sources